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/-
 - Created in 2024 by Gaëtan Serré
-/

/-
 - https://github.com/gaetanserre/SBS-Proofs
-/

import Mathlib

import SBSProofs.Utils

open Classical MeasureTheory

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y)

set_option trace.Meta.Tactic.simp.rewrite true
set_option maxHeartbeats 600000

variable {d : ℕ}  : Set (Vector  d)} [MeasureSpace Ω]

def L2  : Measure Ω) [IsFiniteMeasure μ] := {f : Ω   | Memℒp f 2 μ}

def eigen := {e :    //  i, 0 <= e i}

def f_repr (v : eigen) (e :   Ω  ℝ) (f : Ω  ℝ) (a :   ℝ) := (f = λ x  (∑' i, (v.1 i) * (a i) * (e i x)))  ( x, Summable (λ i  (v.1 i) * (a i) * (e i x)))

/-
  We define a set of functions that depends on a finite measure μ. Each function is representable by a infinite sum.
-/

def H (v : eigen) (e :   Ω  ℝ)  : Measure Ω) [IsFiniteMeasure μ] := {f | f  L2 μ   (a :   ℝ), (f_repr v e f a)  Summable (λ i  (v.1 i) * (a i)^2)}

def set_repr {v : eigen} {e :   Ω  ℝ}  : Measure Ω} [IsFiniteMeasure μ] (f : H v e μ) := {a :    | (f_repr v e f.1 a)  (Summable (λ i  (v.1 i) * (a i)^2))}

lemma set_repr_ne {v : eigen} {e :   Ω  ℝ}  : Measure Ω} [IsFiniteMeasure μ] (f : H v e μ) : (set_repr f).Nonempty := 

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

Set.Nonempty (set_repr f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
left✝: f L2 μ
a:
ha: f_repr v e (f) a Summable fun i => v i * a i ^ 2

intro.intro
Set.Nonempty (set_repr f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

Set.Nonempty (set_repr f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
left✝: f L2 μ
a:
ha: f_repr v e (f) a Summable fun i => v i * a i ^ 2

h
a set_repr f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

Set.Nonempty (set_repr f)

Goals accomplished! 🐙
/- We assume that the the representative of each function in H is unique (property of v and e). -/ axiom set_repr_unique {v : eigen} {e : Ω ℝ} : Measure Ω} [IsFiniteMeasure μ] {f : H v e μ} {a b : ℝ} (ha : a set_repr f) (hb : b set_repr f) : a = b lemma unique_choice {v : eigen} {e : Ω ℝ} : Measure Ω} [IsFiniteMeasure μ] {f : H v e μ} {a : ℝ} (h : a set_repr f) : (set_repr_ne f).some = a := set_repr_unique (set_repr_ne f).some_mem h /- We assume that the product of two representative is summable. -/ axiom product_summable {v : eigen} {e : Ω ℝ} : Measure Ω} [IsFiniteMeasure μ] (f g : H v e μ) : Summable (λ i (v.1 i) * ((set_repr_ne f).some i) * ((set_repr_ne g).some i)) /- We define the multiplication between a real number and a function in H as the pointwise product. We show that the result lies in H. -/ namespace Ring variable {v : eigen} {e : Ω ℝ} : Measure Ω} [IsFiniteMeasure μ] (f g : H v e μ) lemma mul_repr (a : ℝ) : f_repr v e (λ x a * f.1 x) (λ i a * (set_repr_ne f).some i) :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f

_root_.f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f

left
(fun x => a * f x) = fun x => ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
(x : Ω), Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f

left
(fun x => a * f x) = fun x => ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f

left
(fun x => a * f x) = fun x => ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
(x : Ω), Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
x: Ω

left.h
a * f x = ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f

left
(fun x => a * f x) = fun x => ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x

left.h.intro.intro
a * f x = ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f

left
(fun x => a * f x) = fun x => ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x

left.h.intro.intro
a * f x = ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x

g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x

g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x

g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω

h
g x = ∑' (i : ℕ), v i * a * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x

g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω

h
g x = ∑' (i : ℕ), v i * a * h i * e i x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
i:

v i * a * h i * e i x = a * v i * h i * e i x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x

g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x

h
g x = ∑' (i : ℕ), v i * a * h i * e i x
[Meta.Tactic.simp.rewrite] comm:1000, v i * a * h i * e i x ==> a * v i * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x

h
g x = ∑' (i : ℕ), a * v i * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x

h
g x = ∑' (i : ℕ), a * v i * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x

g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x

h
g x = ∑' (i : ℕ), a * v i * h i * e i x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x

(i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x

(i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x

(i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x
i:

a * v i * h i * e i x = a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x

(i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x

g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x
summand_comm: (i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)

h
g x = ∑' (i : ℕ), a * v i * h i * e i x
[Meta.Tactic.simp.rewrite] summand_comm:1000, a * v i * h i * e i x ==> a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x
summand_comm: (i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)

h
g x = ∑' (i : ℕ), a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x
summand_comm: (i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)

h
g x = ∑' (i : ℕ), a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x

g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x
summand_comm: (i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)

h
g x = ∑' (i : ℕ), a * (v i * h i * e i x)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x
summand_comm: (i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)

∑' (i : ℕ), a * (fun i => v i * h i * e i x) i = a * ∑' (i : ℕ), (fun i => v i * h i * e i x) i

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x

g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x
summand_comm: (i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)
const_out: ∑' (i : ℕ), a * (fun i => v i * h i * e i x) i = a * ∑' (i : ℕ), (fun i => v i * h i * e i x) i

h
g x = ∑' (i : ℕ), a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x
summand_comm: (i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)
const_out: ∑' (i : ℕ), a * (fun i => v i * h i * e i x) i = a * ∑' (i : ℕ), (fun i => v i * h i * e i x) i

h
g x = a * ∑' (i : ℕ), (fun i => v i * h i * e i x) i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x
summand_comm: (i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)
const_out: ∑' (i : ℕ), a * (fun i => v i * h i * e i x) i = a * ∑' (i : ℕ), (fun i => v i * h i * e i x) i

h
g x = ∑' (i : ℕ), a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x✝: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
x: Ω
comm: (i : ℕ), v i * a * h i * e i x = a * v i * h i * e i x
summand_comm: (i : ℕ), a * v i * h i * e i x = a * (v i * h i * e i x)
const_out: ∑' (i : ℕ), a * (fun i => v i * h i * e i x) i = a * ∑' (i : ℕ), (fun i => v i * h i * e i x) i

h
g x = a * f x

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f

left
(fun x => a * f x) = fun x => ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * a * h i * e i x

left.h.intro.intro
a * f x = ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f

left
(fun x => a * f x) = fun x => ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * a * h i * e i x

left.h.intro.intro
a * f x = ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
[Meta.Tactic.simp.rewrite] show i, v.1 i * a * h i * e i x = v.1 i * (a * h i) * e i x by intro i; ring:1000, v i * a * h i * e i x ==> v i * (a * h i) * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * a * h i * e i x

left.h.intro.intro
a * f x = ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
[Meta.Tactic.simp.rewrite] show i, v.1 i * a * h i * e i x = v.1 i * (a * h i) * e i x by intro i; ring:1000, v i * a * h i * e i x ==> v i * (a * h i) * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * a * h i * e i x
i:

v i * a * h i * e i x = v i * (a * h i) * e i x
[Meta.Tactic.simp.rewrite] show i, v.1 i * a * h i * e i x = v.1 i * (a * h i) * e i x by intro i; ring:1000, v i * a * h i * e i x ==> v i * (a * h i) * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * a * h i * e i x
i:

v i * a * h i * e i x = v i * (a * h i) * e i x
[Meta.Tactic.simp.rewrite] show i, v.1 i * a * h i * e i x = v.1 i * (a * h i) * e i x by intro i; ring:1000, v i * a * h i * e i x ==> v i * (a * h i) * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * a * h i * e i x

(i : ℕ), v i * a * h i * e i x = v i * (a * h i) * e i x
[Meta.Tactic.simp.rewrite] show i, v.1 i * a * h i * e i x = v.1 i * (a * h i) * e i x by intro i; ring:1000, v i * a * h i * e i x ==> v i * (a * h i) * e i x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * (a * h i) * e i x

left.h.intro.intro
a * f x = ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f

left
(fun x => a * f x) = fun x => ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * (a * h i) * e i x

left.h.intro.intro
a * f x = ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
[Meta.Tactic.simp.rewrite] show i, a * h i = g_h i by intro i; rfl:1000, a * h i ==> g_h i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * (a * h i) * e i x

left.h.intro.intro
a * f x = ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
[Meta.Tactic.simp.rewrite] show i, a * h i = g_h i by intro i; rfl:1000, a * h i ==> g_h i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * (a * h i) * e i x
i:

a * h i = g_h i
[Meta.Tactic.simp.rewrite] show i, a * h i = g_h i by intro i; rfl:1000, a * h i ==> g_h i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * (a * h i) * e i x
i:

a * h i = g_h i
[Meta.Tactic.simp.rewrite] show i, a * h i = g_h i by intro i; rfl:1000, a * h i ==> g_h i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * (a * h i) * e i x

(i : ℕ), a * h i = g_h i
[Meta.Tactic.simp.rewrite] show i, a * h i = g_h i by intro i; rfl:1000, a * h i ==> g_h i

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
x: Ω
right✝¹: Summable fun i => v i * h i ^ 2
f_repr: f = fun x => ∑' (i : ℕ), v i * h i * e i x
right✝: (x : Ω), Summable fun i => v i * h i * e i x
g_eq_tsum: g = fun x => ∑' (i : ℕ), v i * a * h i * e i x
g_x: g x = ∑' (i : ℕ), v i * g_h i * e i x

left.h.intro.intro
a * f x = ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f

left
(fun x => a * f x) = fun x => ∑' (i : ℕ), v i * (fun i => a * Set.Nonempty.some i) i * e i x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
x: Ω

right
Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
x: Ω

right
Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i * e i x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
x: Ω

(fun i => v i * (fun i => a * h i) i * e i x) = fun i => a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
x: Ω

(fun i => v i * (fun i => a * h i) i * e i x) = fun i => a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
x: Ω

(fun i => v i * (fun i => a * h i) i * e i x) = fun i => a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
x: Ω
i:

h
v i * (fun i => a * h i) i * e i x = a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
x: Ω

(fun i => v i * (fun i => a * h i) i * e i x) = fun i => a * (v i * h i * e i x)

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
x: Ω
remove_function: (fun i => v i * (fun i => a * h i) i * e i x) = fun i => a * (v i * h i * e i x)

right
Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
x: Ω
remove_function: (fun i => v i * (fun i => a * h i) i * e i x) = fun i => a * (v i * h i * e i x)

right
Summable fun i => a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
f_repr: h set_repr f
x: Ω
remove_function: (fun i => v i * (fun i => a * h i) i * e i x) = fun i => a * (v i * h i * e i x)

right
Summable fun i => a * (v i * h i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

f_repr v e (fun x => a * f x) fun i => a * Set.Nonempty.some i

Goals accomplished! 🐙
lemma mul_summable (a : ℝ) : Summable (λ i v.1 i * (λ i a * (set_repr_ne f).some i) i ^ 2) :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :

Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:

Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
set_repr: h _root_.set_repr f

Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2

intro
Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2

intro
Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2

(fun i => v i * g_h i ^ 2) = fun i => a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2

(fun i => v i * g_h i ^ 2) = fun i => a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2

(fun i => v i * g_h i ^ 2) = fun i => a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2
i:

h
v i * g_h i ^ 2 = a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2

(fun i => v i * g_h i ^ 2) = fun i => a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2
i:

h
v i * g_h i ^ 2 = a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2
i:

h
v i * g_h i ^ 2 = a ^ 2 * (v i * h i ^ 2)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2
i:

h
v i * (a * h i) ^ 2 = a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2

(fun i => v i * g_h i ^ 2) = fun i => a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2
i:

h
v i * (a * h i) ^ 2 = a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2
i:

h
v i * (a ^ 2 * h i ^ 2) = a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2
i:

h
v i * (a ^ 2 * h i ^ 2) = a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2

(fun i => v i * g_h i ^ 2) = fun i => a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2
i:

h
v i * (a ^ 2 * h i ^ 2) = a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2
i:

h
v i * (a ^ 2 * h i ^ 2) = a ^ 2 * (v i * h i ^ 2)

Goals accomplished! 🐙

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2
comm_fun: (fun i => v i * g_h i ^ 2) = fun i => a ^ 2 * (v i * h i ^ 2)

intro
Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2
comm_fun: (fun i => v i * g_h i ^ 2) = fun i => a ^ 2 * (v i * h i ^ 2)

intro
Summable fun i => a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:
left✝: f_repr v e (f) h
h_summable: Summable fun i => v i * h i ^ 2
comm_fun: (fun i => v i * g_h i ^ 2) = fun i => a ^ 2 * (v i * h i ^ 2)

intro
Summable fun i => a ^ 2 * (v i * h i ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

Summable fun i => v i * (fun i => a * Set.Nonempty.some i) i ^ 2

Goals accomplished! 🐙
lemma mul_in_H (a : ℝ) : (λ x a * f.1 x) (H v e μ) :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

(fun x => a * f x) H v e μ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω

(fun x => a * f x) H v e μ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

(fun x => a * f x) H v e μ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
g_in_L2: g L2 μ

(fun x => a * f x) H v e μ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

(fun x => a * f x) H v e μ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
g_in_L2: g L2 μ
h:= Set.Nonempty.some :

(fun x => a * f x) H v e μ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

(fun x => a * f x) H v e μ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g✝: (H v e μ)
a:
g:= fun x => a * f x: Ω
g_in_L2: g L2 μ
h:= Set.Nonempty.some :
g_h:= fun i => a * h i:

(fun x => a * f x) H v e μ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

(fun x => a * f x) H v e μ

Goals accomplished! 🐙
instance : HMul (H v e μ) (H v e μ) where hMul := λ a f λ x a * f.1 x, mul_in_H f a⟩ instance : HSMul (H v e μ) (H v e μ) where hSMul := λ r f r * f end Ring /- We define the sum between two functions in H as the pointwise sum. We show that the result lies in H. We also define the 0 function of H. We show several properties on the addition. -/ namespace Group variable {v : eigen} {e : Ω ℝ} : Measure Ω} [IsFiniteMeasure μ] (f g : H v e μ) lemma add_in_L2 : (λ x f.1 x + g.1 x) L2 μ := Memℒp.add (f.2.1) (g.2.1) lemma add_summable : Summable (λ i v.1 i * ((set_repr_ne f).some i + (set_repr_ne g).some i)^2) :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Summable fun i => v i * (Set.Nonempty.some i + Set.Nonempty.some i) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :

Summable fun i => v i * (Set.Nonempty.some i + Set.Nonempty.some i) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Summable fun i => v i * (Set.Nonempty.some i + Set.Nonempty.some i) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :

Summable fun i => v i * (Set.Nonempty.some i + Set.Nonempty.some i) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Summable fun i => v i * (Set.Nonempty.some i + Set.Nonempty.some i) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :

Summable fun i => v i * (Set.Nonempty.some i + Set.Nonempty.some i) ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :

(fun i => v i * (a_f i + a_g i) ^ 2) = fun i => v i * a_f i * a_f i + 2 * (v i * a_f i * a_g i) + v i * a_g i * a_g i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :

(fun i => v i * (a_f i + a_g i) ^ 2) = fun i => v i * a_f i * a_f i + 2 * (v i * a_f i * a_g i) + v i * a_g i * a_g i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :

(fun i => v i * (a_f i + a_g i) ^ 2) = fun i => v i * a_f i * a_f i + 2 * (v i * a_f i * a_g i) + v i * a_g i * a_g i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
i:

h
v i * (a_f i + a_g i) ^ 2 = v i * a_f i * a_f i + 2 * (v i * a_f i * a_g i) + v i * a_g i * a_g i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :

(fun i => v i * (a_f i + a_g i) ^ 2) = fun i => v i * a_f i * a_f i + 2 * (v i * a_f i * a_g i) + v i * a_g i * a_g i

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Summable fun i => v i * (Set.Nonempty.some i + Set.Nonempty.some i) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
f_eq: (fun i => v i * (a_f i + a_g i) ^ 2) = fun i => v i * a_f i * a_f i + 2 * (v i * a_f i * a_g i) + v i * a_g i * a_g i

Summable fun i => v i * (Set.Nonempty.some i + Set.Nonempty.some i) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
f_eq: (fun i => v i * (a_f i + a_g i) ^ 2) = fun i => v i * a_f i * a_f i + 2 * (v i * a_f i * a_g i) + v i * a_g i * a_g i

Summable fun i => v i * a_f i * a_f i + 2 * (v i * a_f i * a_g i) + v i * a_g i * a_g i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
f_eq: (fun i => v i * (a_f i + a_g i) ^ 2) = fun i => v i * a_f i * a_f i + 2 * (v i * a_f i * a_g i) + v i * a_g i * a_g i

Summable fun i => v i * a_f i * a_f i + 2 * (v i * a_f i * a_g i) + v i * a_g i * a_g i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Summable fun i => v i * (Set.Nonempty.some i + Set.Nonempty.some i) ^ 2

Goals accomplished! 🐙
lemma add_repr : f_repr v e (λ x f.1 x + g.1 x) (λ i (set_repr_ne f).some i + (set_repr_ne g).some i) :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :

f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :

f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2

intro
f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2

intro.intro
f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2

intro.intro
f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2

(x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
x: Ω
i:

v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
x: Ω
i:

v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2

(x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x

intro.intro.left
(fun x => f x + g x) = fun x => ∑' (i : ℕ), v i * (fun i => Set.Nonempty.some i + Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
(x : Ω), Summable fun i => v i * (fun i => Set.Nonempty.some i + Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x

intro.intro.left
(fun x => f x + g x) = fun x => ∑' (i : ℕ), v i * (fun i => Set.Nonempty.some i + Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x

intro.intro.left
(fun x => f x + g x) = fun x => ∑' (i : ℕ), v i * (fun i => Set.Nonempty.some i + Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
(x : Ω), Summable fun i => v i * (fun i => Set.Nonempty.some i + Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.left.h
f x + g x = ∑' (i : ℕ), v i * (fun i => Set.Nonempty.some i + Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x

intro.intro.left
(fun x => f x + g x) = fun x => ∑' (i : ℕ), v i * (fun i => Set.Nonempty.some i + Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.left.h
f x + g x = ∑' (i : ℕ), v i * (Set.Nonempty.some i + Set.Nonempty.some i) * e i x
[Meta.Tactic.simp.rewrite] summand_distrib x:1000, v i * (Set.Nonempty.some i + Set.Nonempty.some i) * e i x ==> v i * a_f i * e i x + v i * a_g i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.left.h
f x + g x = ∑' (i : ℕ), (v i * a_f i * e i x + v i * a_g i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.left.h
f x + g x = ∑' (i : ℕ), (v i * a_f i * e i x + v i * a_g i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x

intro.intro.left
(fun x => f x + g x) = fun x => ∑' (i : ℕ), v i * (fun i => Set.Nonempty.some i + Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.left.h
f x + g x = ∑' (i : ℕ), (v i * a_f i * e i x + v i * a_g i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.left.h
f x + g x = ∑' (b : ℕ), v b * Set.Nonempty.some b * e b x + ∑' (b : ℕ), v b * Set.Nonempty.some b * e b x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.left.h
f x + g x = ∑' (i : ℕ), (v i * a_f i * e i x + v i * a_g i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.left.h
f x + g x = f x + ∑' (b : ℕ), v b * Set.Nonempty.some b * e b x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.left.h
f x + g x = ∑' (i : ℕ), (v i * a_f i * e i x + v i * a_g i * e i x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.left.h
f x + g x = f x + g x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.left.h
f x + g x = ∑' (i : ℕ), (v i * a_f i * e i x + v i * a_g i * e i x)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.right
Summable fun i => v i * (fun i => Set.Nonempty.some i + Set.Nonempty.some i) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.right
Summable fun i => v i * (Set.Nonempty.some i + Set.Nonempty.some i) * e i x
[Meta.Tactic.simp.rewrite] summand_distrib x:1000, v i * (Set.Nonempty.some i + Set.Nonempty.some i) * e i x ==> v i * a_f i * e i x + v i * a_g i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.right
Summable fun i => v i * a_f i * e i x + v i * a_g i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
af_repr: f_repr v e (f) (Set.Nonempty.some )
right✝¹: Summable fun i => v i * Set.Nonempty.some i ^ 2
ag_repr: f_repr v e (g) (Set.Nonempty.some )
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
summand_distrib: (x : Ω) (i : ℕ), v i * (a_f i + a_g i) * e i x = v i * a_f i * e i x + v i * a_g i * e i x
x: Ω

intro.intro.right
Summable fun i => v i * a_f i * e i x + v i * a_g i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e (fun x => f x + g x) fun i => Set.Nonempty.some i + Set.Nonempty.some i

Goals accomplished! 🐙
/- We define the 0 of H as pointwise 0 function. We show that it lies in H. -/ def zero : Ω := λ _ 0 lemma zero_repr : f_repr v e zero (λ _ 0) :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e zero fun x => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

f_repr v e zero fun x => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e zero fun x => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

left
zero = fun x => ∑' (i : ℕ), v i * (fun x => 0) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
(x : Ω), Summable fun i => v i * (fun x => 0) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e zero fun x => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

left
zero = fun x => ∑' (i : ℕ), v i * (fun x => 0) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

left
zero = fun x => ∑' (i : ℕ), v i * (fun x => 0) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
(x : Ω), Summable fun i => v i * (fun x => 0) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

left.h
zero x = ∑' (i : ℕ), v i * (fun x => 0) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

left
zero = fun x => ∑' (i : ℕ), v i * (fun x => 0) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

left.h
zero x = ∑' (i : ℕ), v i * (fun x => 0) i * e i x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

(i : ℕ), v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

(i : ℕ), v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

(i : ℕ), v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
i:

v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

(i : ℕ), v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
i:

v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
i:

v i * a i * e i x = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
i:

v i * 0 * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

(i : ℕ), v i * a i * e i x = 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

left
zero = fun x => ∑' (i : ℕ), v i * (fun x => 0) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
summand_zero: (i : ℕ), v i * a i * e i x = 0

left.h
zero x = ∑' (i : ℕ), v i * 0 * e i x
[Meta.Tactic.simp.rewrite] summand_zero:1000, v i * 0 * e i x ==> 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
summand_zero: (i : ℕ), v i * a i * e i x = 0

left.h
zero x = ∑' (i : ℕ), v i * 0 * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
summand_zero: (i : ℕ), v i * a i * e i x = 0

left.h
zero x = ∑' (i : ℕ), 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
summand_zero: (i : ℕ), v i * a i * e i x = 0

left.h
zero x = ∑' (i : ℕ), v i * (fun x => 0) i * e i x
[Meta.Tactic.simp.rewrite] @tsum_zero:1000, ∑' (i : ℕ), 0 ==> 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
summand_zero: (i : ℕ), v i * a i * e i x = 0

left.h
zero x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
summand_zero: (i : ℕ), v i * a i * e i x = 0

left.h
zero x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

left
zero = fun x => ∑' (i : ℕ), v i * (fun x => 0) i * e i x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e zero fun x => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

right
Summable fun i => v i * (fun x => 0) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e zero fun x => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

right
Summable fun i => v i * (fun x => 0) i * e i x
Warning: unused variable `i` [linter.unusedVariables]
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

right
Summable fun i => v i * (fun x => 0) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

right
Summable fun i => v i * (fun x => 0) i * e i x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

(fun i => v i * a i * e i x) = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

(fun i => v i * a i * e i x) = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

(fun i => v i * a i * e i x) = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
i:

h
v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

(fun i => v i * a i * e i x) = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
i:

h
v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
i:

h
v i * a i * e i x = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
i:

h
v i * 0 * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω

(fun i => v i * a i * e i x) = fun i => 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e zero fun x => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
null_function: (fun i => v i * a i * e i x) = fun i => 0

right
Summable fun i => v i * (fun x => 0) i * e i x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
null_function: (fun i => v i * a i * e i x) = fun i => 0

right
Summable fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
x: Ω
null_function: (fun i => v i * a i * e i x) = fun i => 0

right
Summable fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f_repr v e zero fun x => 0

Goals accomplished! 🐙
lemma zero_summable : Summable (λ i (v.1 i) * (0 : ℝ)^2) :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Summable fun i => v i * 0 ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Summable fun i => v i * 0 ^ 2
Warning: unused variable `i` [linter.unusedVariables]
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Summable fun i => v i * 0 ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Summable fun i => v i * 0 ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(fun i => v i * 0 ^ 2) = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(fun i => v i * 0 ^ 2) = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(fun i => v i * 0 ^ 2) = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
i:

h
v i * 0 ^ 2 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(fun i => v i * 0 ^ 2) = fun i => 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Summable fun i => v i * 0 ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
zero_fun: (fun i => v i * 0 ^ 2) = fun i => 0

Summable fun i => v i * 0 ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
zero_fun: (fun i => v i * 0 ^ 2) = fun i => 0

Summable fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
zero_fun: (fun i => v i * 0 ^ 2) = fun i => 0

Summable fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Summable fun i => v i * 0 ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
zero_fun: (fun i => v i * 0 ^ 2) = fun i => 0

Summable fun i => 0
Warning: unused variable `i` [linter.unusedVariables]
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
zero_fun: (fun i => v i * 0 ^ 2) = fun i => 0

Summable fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
zero_fun: (fun i => v i * 0 ^ 2) = fun i => 0

Summable fun i => 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
zero_fun: (fun i => v i * 0 ^ 2) = fun i => 0

b , (fun i => 0) b = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
zero_fun: (fun i => v i * 0 ^ 2) = fun i => 0

b , (fun i => 0) b = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
zero_fun: (fun i => v i * 0 ^ 2) = fun i => 0

b , (fun i => 0) b = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
zero_fun: (fun i => v i * 0 ^ 2) = fun i => 0

b , (fun i => 0) b = 0
Warning: unused variable `b_not_in` [linter.unusedVariables]
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
zero_fun: (fun i => v i * 0 ^ 2) = fun i => 0
b:
b_not_in: b

(fun i => 0) b = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
zero_fun: (fun i => v i * 0 ^ 2) = fun i => 0

b , (fun i => 0) b = 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Summable fun i => v i * 0 ^ 2

Goals accomplished! 🐙
lemma zero_in_H : zero L2 μ (a : ℝ), (f_repr v e zero a) Summable (λ i (v.1 i) * (a i)^2) := ⟨memℒp_const 0, (λ _ 0), zero_repr, zero_summable⟩ instance : Zero (H v e μ) where zero := ⟨zero, zero_in_H⟩ lemma zero_unique_repr : (set_repr_ne (0 : H v e μ)).some = (λ i 0) :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Set.Nonempty.some = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

Set.Nonempty.some = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Set.Nonempty.some = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

Set.Nonempty.some = fun i => 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

a set_repr 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

a set_repr 0
--have tmp := zero_repr
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

Summable fun i => v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

a set_repr 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

Summable fun i => v i * a i ^ 2
Warning: unused variable `i` [linter.unusedVariables]
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

Summable fun i => v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

Summable fun i => v i * a i ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

(fun i => v i * a i ^ 2) = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

(fun i => v i * a i ^ 2) = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

(fun i => v i * a i ^ 2) = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
i:

h
v i * a i ^ 2 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

(fun i => v i * a i ^ 2) = fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
i:

h
v i * a i ^ 2 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
i:

h
v i * a i ^ 2 = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
i:

h
v i * 0 ^ 2 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

(fun i => v i * a i ^ 2) = fun i => 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

a set_repr 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
null_function: (fun i => v i * a i ^ 2) = fun i => 0

Summable fun i => v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
null_function: (fun i => v i * a i ^ 2) = fun i => 0

Summable fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:
null_function: (fun i => v i * a i ^ 2) = fun i => 0

Summable fun i => 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= fun x => 0:

a set_repr 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

Set.Nonempty.some = fun i => 0

Goals accomplished! 🐙
lemma add_in_H : (λ x f.1 x + g.1 x) H v e μ :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(fun x => f x + g x) H v e μ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
h:= fun i => Set.Nonempty.some i + Set.Nonempty.some i:

(fun x => f x + g x) H v e μ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(fun x => f x + g x) H v e μ

Goals accomplished! 🐙
instance : HAdd (H v e μ) (H v e μ) (H v e μ) where hAdd := λ f g ⟨(λ x f.1 x + g.1 x), add_in_H f g⟩ instance : HSub (H v e μ) (H v e μ) (H v e μ) where hSub := λ f g f + (-1 : ℝ) * g instance : Neg (H v e μ) where neg := λ f (-1 : ℝ) * f lemma H_add_assoc (a b c : H v e μ) : a + b + c = a + (b + c) :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a + b + c = a + (b + c)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a + b + c) x = (a + (b + c)) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a + b + c = a + (b + c)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a + b + c) x = (a + (b + c)) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a + b + c) x = (a + (b + c)) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
a x + b x + c x = (a + (b + c)) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a + b + c = a + (b + c)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
a x + b x + c x = (a + (b + c)) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
a x + b x + c x = (a + (b + c)) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
a x + b x + c x = a x + (b x + c x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a + b + c = a + (b + c)

Goals accomplished! 🐙
lemma H_add_comm (a b : H v e μ) : a + b = b + a :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

a + b = b + a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
x: Ω

a.h
(a + b) x = (b + a) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

a + b = b + a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
x: Ω

a.h
(a + b) x = (b + a) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
x: Ω

a.h
(a + b) x = (b + a) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
x: Ω

a.h
a x + b x = (b + a) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

a + b = b + a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
x: Ω

a.h
a x + b x = (b + a) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
x: Ω

a.h
a x + b x = (b + a) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
x: Ω

a.h
a x + b x = b x + a x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

a + b = b + a

Goals accomplished! 🐙
lemma H_add_zero (a : H v e μ) : a + 0 = a :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a + 0 = a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
(a + 0) x = a x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a + 0 = a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
(a + 0) x = a x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
(a + 0) x = a x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
a x + 0 x = a x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a + 0 = a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
a x + 0 x = a x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
a x + 0 x = a x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
a x + 0 = a x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a + 0 = a

Goals accomplished! 🐙
lemma H_zero_add (a : H v e μ) : 0 + a = a :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

0 + a = a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

0 + a = a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a + 0 = a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a + 0 = a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

0 + a = a

Goals accomplished! 🐙
lemma H_add_left_neg (a : H v e μ) : -a + a = 0 :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

-a + a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

-a + a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

-a + a = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

-1 * a + a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

-a + a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
(-1 * a + a) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

-a + a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
(-1 * a + a) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
(-1 * a + a) x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
(-1 * a) x + a x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

-a + a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
(-1 * a) x + a x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
(-1 * a) x + a x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
-1 * a x + a x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

-a + a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
-1 * a x + a x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
-1 * a x + a x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
-1 * a x + a x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

-a + a = 0

Goals accomplished! 🐙
end Group /- We define a function : H × H → ℝ. The purpose of the following is to prove that H endowed with this function is a inner product space. -/ noncomputable def H_inner {v : eigen} {e : Ω ℝ} : Measure Ω} [IsFiniteMeasure μ] (f g : H v e μ) : := ∑' i, (v.1 i) * ((set_repr_ne f).some i) * ((set_repr_ne g).some i) /- - We show properties on the inner product of H and the induced norm. -/ namespace Inner open Ring Group variable {v : eigen} {e : Ω ℝ} : Measure Ω} [IsFiniteMeasure μ] (f g : H v e μ) noncomputable instance : Inner (H v e μ) where inner := H_inner noncomputable instance : Norm (H v e μ) where norm := λ f Real.sqrt (inner f f) noncomputable instance : Dist (H v e μ) where dist := λ f g norm (f - g) lemma inner_mul_left (a : ℝ) : inner (a * f) g = a * inner f g :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

a * f, g_ℝ = a * f, g_ℝ

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

H_inner (a * f) g = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

∑' (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

∑' (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

∑' (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * f, g_ℝ

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

(i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

(i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

(i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
i:

v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)

(i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
comm_summand: (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i

∑' (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * f, g_ℝ

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
comm_summand: (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i

(i : ℕ), a * v i * h i * Set.Nonempty.some i = a * (fun i => v i * h i * Set.Nonempty.some i) i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
comm_summand: (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i

(i : ℕ), a * v i * h i * Set.Nonempty.some i = a * (fun i => v i * h i * Set.Nonempty.some i) i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
comm_summand: (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i

(i : ℕ), a * v i * h i * Set.Nonempty.some i = a * (fun i => v i * h i * Set.Nonempty.some i) i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
comm_summand: (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i
i:

a * v i * h i * Set.Nonempty.some i = a * (fun i => v i * h i * Set.Nonempty.some i) i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
comm_summand: (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i

(i : ℕ), a * v i * h i * Set.Nonempty.some i = a * (fun i => v i * h i * Set.Nonempty.some i) i

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

a * f, g_ℝ = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
comm_summand: (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i
lambda_comm: (i : ℕ), a * v i * h i * Set.Nonempty.some i = a * (fun i => v i * h i * Set.Nonempty.some i) i

∑' (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * f, g_ℝ
[Meta.Tactic.simp.rewrite] comm_summand:1000, v i * h_af i * Set.Nonempty.some i ==> a * v i * h i * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
comm_summand: (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i
lambda_comm: (i : ℕ), a * v i * h i * Set.Nonempty.some i = a * (fun i => v i * h i * Set.Nonempty.some i) i

∑' (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
comm_summand: (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i
lambda_comm: (i : ℕ), a * v i * h i * Set.Nonempty.some i = a * (fun i => v i * h i * Set.Nonempty.some i) i

∑' (i : ℕ), a * v i * h i * Set.Nonempty.some i = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
comm_summand: (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i
lambda_comm: (i : ℕ), a * v i * h i * Set.Nonempty.some i = a * (fun i => v i * h i * Set.Nonempty.some i) i

∑' (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * f, g_ℝ
[Meta.Tactic.simp.rewrite] lambda_comm:1000, a * v i * h i * Set.Nonempty.some i ==> a * (fun i => v i * h i * Set.Nonempty.some i) i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
comm_summand: (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i
lambda_comm: (i : ℕ), a * v i * h i * Set.Nonempty.some i = a * (fun i => v i * h i * Set.Nonempty.some i) i

∑' (i : ℕ), a * (v i * h i * Set.Nonempty.some i) = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:
h:= Set.Nonempty.some :
h_af:= fun i => a * h i:
h_af_in: h_af set_repr (a * f)
comm_summand: (i : ℕ), v i * h_af i * Set.Nonempty.some i = a * v i * h i * Set.Nonempty.some i
lambda_comm: (i : ℕ), a * v i * h i * Set.Nonempty.some i = a * (fun i => v i * h i * Set.Nonempty.some i) i

∑' (i : ℕ), a * (v i * h i * Set.Nonempty.some i) = a * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:

a * f, g_ℝ = a * f, g_ℝ

Goals accomplished! 🐙
lemma H_inner_add_left (h : H v e μ) : (inner (f + g) h : ℝ) = inner f h + inner g h :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

H_inner (f + g) h = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (i : ℕ), v i * a_fg i * Set.Nonempty.some i = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (i : ℕ), v i * a_fg i * Set.Nonempty.some i = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (i : ℕ), v i * a_fg i * Set.Nonempty.some i = f, h_ℝ + g, h_ℝ
[Meta.Tactic.simp.rewrite] show i, v.1 i * a_fg i * a_h i = v.1 i * (a_f i + a_g i) * a_h i by intro i; rfl:1000, v i * a_fg i * Set.Nonempty.some i ==> v i * (a_f i + a_g i) * a_h i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (i : ℕ), v i * a_fg i * Set.Nonempty.some i = f, h_ℝ + g, h_ℝ
[Meta.Tactic.simp.rewrite] show i, v.1 i * a_fg i * a_h i = v.1 i * (a_f i + a_g i) * a_h i by intro i; rfl:1000, v i * a_fg i * Set.Nonempty.some i ==> v i * (a_f i + a_g i) * a_h i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)
i:

v i * a_fg i * a_h i = v i * (a_f i + a_g i) * a_h i
[Meta.Tactic.simp.rewrite] show i, v.1 i * a_fg i * a_h i = v.1 i * (a_f i + a_g i) * a_h i by intro i; rfl:1000, v i * a_fg i * Set.Nonempty.some i ==> v i * (a_f i + a_g i) * a_h i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)
i:

v i * a_fg i * a_h i = v i * (a_f i + a_g i) * a_h i
[Meta.Tactic.simp.rewrite] show i, v.1 i * a_fg i * a_h i = v.1 i * (a_f i + a_g i) * a_h i by intro i; rfl:1000, v i * a_fg i * Set.Nonempty.some i ==> v i * (a_f i + a_g i) * a_h i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

(i : ℕ), v i * a_fg i * a_h i = v i * (a_f i + a_g i) * a_h i
[Meta.Tactic.simp.rewrite] show i, v.1 i * a_fg i * a_h i = v.1 i * (a_f i + a_g i) * a_h i by intro i; rfl:1000, v i * a_fg i * Set.Nonempty.some i ==> v i * (a_f i + a_g i) * a_h i

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (i : ℕ), v i * (a_f i + a_g i) * a_h i = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (i : ℕ), v i * (a_f i + a_g i) * a_h i = f, h_ℝ + g, h_ℝ
[Meta.Tactic.simp.rewrite] show i, v.1 i * (a_f i + a_g i) * a_h i = v.1 i * a_f i * a_h i + v.1 i * a_g i * a_h i by intro i; ring:1000, v i * (a_f i + a_g i) * a_h i ==> v i * a_f i * a_h i + v i * a_g i * a_h i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (i : ℕ), v i * (a_f i + a_g i) * a_h i = f, h_ℝ + g, h_ℝ
[Meta.Tactic.simp.rewrite] show i, v.1 i * (a_f i + a_g i) * a_h i = v.1 i * a_f i * a_h i + v.1 i * a_g i * a_h i by intro i; ring:1000, v i * (a_f i + a_g i) * a_h i ==> v i * a_f i * a_h i + v i * a_g i * a_h i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)
i:

v i * (a_f i + a_g i) * a_h i = v i * a_f i * a_h i + v i * a_g i * a_h i
[Meta.Tactic.simp.rewrite] show i, v.1 i * (a_f i + a_g i) * a_h i = v.1 i * a_f i * a_h i + v.1 i * a_g i * a_h i by intro i; ring:1000, v i * (a_f i + a_g i) * a_h i ==> v i * a_f i * a_h i + v i * a_g i * a_h i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)
i:

v i * (a_f i + a_g i) * a_h i = v i * a_f i * a_h i + v i * a_g i * a_h i
[Meta.Tactic.simp.rewrite] show i, v.1 i * (a_f i + a_g i) * a_h i = v.1 i * a_f i * a_h i + v.1 i * a_g i * a_h i by intro i; ring:1000, v i * (a_f i + a_g i) * a_h i ==> v i * a_f i * a_h i + v i * a_g i * a_h i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

(i : ℕ), v i * (a_f i + a_g i) * a_h i = v i * a_f i * a_h i + v i * a_g i * a_h i
[Meta.Tactic.simp.rewrite] show i, v.1 i * (a_f i + a_g i) * a_h i = v.1 i * a_f i * a_h i + v.1 i * a_g i * a_h i by intro i; ring:1000, v i * (a_f i + a_g i) * a_h i ==> v i * a_f i * a_h i + v i * a_g i * a_h i

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (i : ℕ), (v i * a_f i * a_h i + v i * a_g i * a_h i) = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (i : ℕ), (v i * a_f i * a_h i + v i * a_g i * a_h i) = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (b : ℕ), v b * Set.Nonempty.some b * Set.Nonempty.some b + ∑' (b : ℕ), v b * Set.Nonempty.some b * Set.Nonempty.some b = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_h:= Set.Nonempty.some :
a_fg:= fun i => a_f i + a_g i:
a_fg_repr: a_fg set_repr (f + g)

∑' (b : ℕ), v b * Set.Nonempty.some b * Set.Nonempty.some b + ∑' (b : ℕ), v b * Set.Nonempty.some b * Set.Nonempty.some b = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ

Goals accomplished! 🐙
lemma inner_symmetric : (inner f g : ℝ) = inner g f :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ = g, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ = g, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ = g, f_ℝ

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

H_inner f g = g, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ = g, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = g, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ = g, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = g, f_ℝ

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
i:

v i * Set.Nonempty.some i * Set.Nonempty.some i = v i * Set.Nonempty.some i * Set.Nonempty.some i

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ = g, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
comm: (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = v i * Set.Nonempty.some i * Set.Nonempty.some i

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = g, f_ℝ
[Meta.Tactic.simp.rewrite] comm:1000, v i * Set.Nonempty.some i * Set.Nonempty.some i ==> v i * Set.Nonempty.some i * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
comm: (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = v i * Set.Nonempty.some i * Set.Nonempty.some i

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = g, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
comm: (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = v i * Set.Nonempty.some i * Set.Nonempty.some i

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = g, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ = g, f_ℝ

Goals accomplished! 🐙
lemma inner_nonneg : (0 : ℝ) <= inner f f :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f, f_ℝ

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 H_inner f f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :

0 ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :

0 ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :

(i : ℕ), v i * a i * a i = v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :

(i : ℕ), v i * a i * a i = v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :

(i : ℕ), v i * a i * a i = v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
i:

v i * a i * a i = v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :

(i : ℕ), v i * a i * a i = v i * a i ^ 2

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq: (i : ℕ), v i * a i * a i = v i * a i ^ 2

0 ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i
[Meta.Tactic.simp.rewrite] sq:1000, v i * Set.Nonempty.some i * Set.Nonempty.some i ==> v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq: (i : ℕ), v i * a i * a i = v i * a i ^ 2

0 ∑' (i : ℕ), v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq: (i : ℕ), v i * a i * a i = v i * a i ^ 2

0 ∑' (i : ℕ), v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq: (i : ℕ), v i * a i * a i = v i * a i ^ 2

0 ∑' (i : ℕ), v i * a i ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq: (i : ℕ), v i * a i * a i = v i * a i ^ 2

(i : ℕ), 0 v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq: (i : ℕ), v i * a i * a i = v i * a i ^ 2

(i : ℕ), 0 v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq: (i : ℕ), v i * a i * a i = v i * a i ^ 2

(i : ℕ), 0 v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq: (i : ℕ), v i * a i * a i = v i * a i ^ 2
i:

0 v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq: (i : ℕ), v i * a i * a i = v i * a i ^ 2

(i : ℕ), 0 v i * a i ^ 2

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f, f_ℝ

Goals accomplished! 🐙
lemma H_norm_nonneg : (0 : ℝ) <= norm f :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 √⟪f, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

0 f

Goals accomplished! 🐙
lemma inner_zero_eq_zero : inner f 0 = (0 : ℝ) :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, 0_ℝ = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, 0_ℝ = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, 0_ℝ = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

H_inner f 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, 0_ℝ = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, 0_ℝ = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * (fun i => 0) i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * (fun i => 0) i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, 0_ℝ = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * (fun i => 0) i = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(i : ℕ), v i * Set.Nonempty.some i * (fun i => 0) i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(i : ℕ), v i * Set.Nonempty.some i * (fun i => 0) i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(i : ℕ), v i * Set.Nonempty.some i * (fun i => 0) i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
i:

v i * Set.Nonempty.some i * (fun i => 0) i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(i : ℕ), v i * Set.Nonempty.some i * (fun i => 0) i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
i:

v i * Set.Nonempty.some i * (fun i => 0) i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
i:

v i * Set.Nonempty.some i * (fun i => 0) i = 0
Warning: unused variable `i` [linter.unusedVariables]
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
i:

v i * Set.Nonempty.some i * (fun i => 0) i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
i:

v i * Set.Nonempty.some i * (fun i => 0) i = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
i:

v i * Set.Nonempty.some i * 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(i : ℕ), v i * Set.Nonempty.some i * (fun i => 0) i = 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, 0_ℝ = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
summand_eq_zero: (i : ℕ), v i * Set.Nonempty.some i * (fun i => 0) i = 0

∑' (i : ℕ), v i * Set.Nonempty.some i * 0 = 0
[Meta.Tactic.simp.rewrite] summand_eq_zero:1000, v i * Set.Nonempty.some i * 0 ==> 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
summand_eq_zero: (i : ℕ), v i * Set.Nonempty.some i * (fun i => 0) i = 0

∑' (i : ℕ), 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
summand_eq_zero: (i : ℕ), v i * Set.Nonempty.some i * (fun i => 0) i = 0

∑' (i : ℕ), 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, 0_ℝ = 0

Goals accomplished! 🐙
lemma null_inner_imp_null_f : inner f f = (0 : ℝ) f = 0 :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
inner_eq_0: f, f_ℝ = 0

f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
inner_eq_0: f, f_ℝ = 0

f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
inner_eq_0: f, f_ℝ = 0

f = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
inner_eq_0: H_inner f f = 0

f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
inner_eq_0: H_inner f f = 0

f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
inner_eq_0: ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = 0

f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
inner_eq_0: ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = 0
a:= Set.Nonempty.some :

f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
inner_eq_0: ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = 0
a:= Set.Nonempty.some :

f = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
inner_eq_0: ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = 0
a:= Set.Nonempty.some :
i:

v i * a i * a i = v i * a i ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
inner_eq_0: ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = 0
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2

f = 0
[Meta.Tactic.simp.rewrite] sq_summand:1000, v i * Set.Nonempty.some i * Set.Nonempty.some i ==> v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0

f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0

f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0

f = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0

(i : ℕ), 0 v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0

(i : ℕ), 0 v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0

(i : ℕ), 0 v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
i:

0 v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0

(i : ℕ), 0 v i * a i ^ 2

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2

f = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2

Summable fun i => v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2

Summable fun i => v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2

Summable fun i => v i * a i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2

Summable fun i => v i * a i ^ 2
[Meta.Tactic.simp.rewrite] sq_summand:1000, v i * a i ^ 2 ==> v i * a i * a i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2

Summable fun i => v i * a i * a i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2

Summable fun i => v i * a i * a i
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2

Summable fun i => v i * a i ^ 2

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0

f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0

f = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0

(i : ℕ), v i * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0

(i : ℕ), v i * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0

(i : ℕ), v i * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:

v i * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0

(i : ℕ), v i * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:
x✝: v i = 0 a i ^ 2 = 0

v i * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:
hv: v i = 0

inl
v i * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:
hv: v i = 0

inl
v i * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:
hv: v i = 0

inl
0 * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:
hv: v i = 0

inl
0 * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:
hv: v i = 0

inl
v i * a i = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:
x✝: v i = 0 a i ^ 2 = 0

v i * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:
ha: a i ^ 2 = 0

inr
v i * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:
ha: a i ^ 2 = 0

inr
v i * a i = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:
ha: a i ^ 2 = 0

inr
v i * 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:
ha: a i ^ 2 = 0

inr
v i * 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
i:
ha: a i ^ 2 = 0

inr
v i * a i = 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω

a.h
f x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω

a.h
f x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω

a.h
f x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω

a.h
f x = zero x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω

a.h
f x = zero x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω

a.h
f x = zero x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω

a.h
f x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω

a.h
f x = 0
Warning: unused variable `ha_s` [linter.unusedVariables]
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω

a.h
f x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x

a.h.intro.intro
f x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x

a.h.intro.intro
f x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x

a.h.intro.intro
(fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x) x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x

a.h.intro.intro
f x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x

a.h.intro.intro
(fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x) x = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x

a.h.intro.intro
(fun x => ∑' (i : ℕ), v i * a i * e i x) x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x

a.h.intro.intro
(fun x => ∑' (i : ℕ), v i * a i * e i x) x = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x

(i : ℕ), v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x

(i : ℕ), v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x

(i : ℕ), v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x
i:

v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x

(i : ℕ), v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x
i:

v i * a i * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x
i:

0 * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x
i:

0 * e i x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x

(i : ℕ), v i * a i * e i x = 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x
summand_eq_0: (i : ℕ), v i * a i * e i x = 0

a.h.intro.intro
∑' (i : ℕ), v i * a i * e i x = 0
[Meta.Tactic.simp.rewrite] summand_eq_0:1000, v i * a i * e i x ==> 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x
summand_eq_0: (i : ℕ), v i * a i * e i x = 0

a.h.intro.intro
∑' (i : ℕ), 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
a:= Set.Nonempty.some :
sq_summand: (i : ℕ), v i * a i * a i = v i * a i ^ 2
inner_eq_0: ∑' (i : ℕ), v i * a i ^ 2 = 0
summand_nonneg: (i : ℕ), 0 v i * a i ^ 2
summand_summable: Summable fun i => v i * a i ^ 2
summand_eq_zero: (i : ℕ), v i * a i ^ 2 = 0
mul_v_a_eq_0: (i : ℕ), v i * a i = 0
x: Ω
right✝: Summable fun i => v i * Set.Nonempty.some i ^ 2
ha_r: f = fun x => ∑' (i : ℕ), v i * Set.Nonempty.some i * e i x
ha_s: (x : Ω), Summable fun i => v i * Set.Nonempty.some i * e i x
summand_eq_0: (i : ℕ), v i * a i * e i x = 0

a.h.intro.intro
∑' (i : ℕ), 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = 0 f = 0

Goals accomplished! 🐙
lemma inner_eq_sq_norm : inner f f = (norm f)^2 :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = f ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = f ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = f ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = √⟪f, f_ℝ ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = f ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = √⟪f, f_ℝ ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, f_ℝ = f, f_ℝ

Goals accomplished! 🐙
lemma distrib_H_norm (t : ℝ) : f + t*g‖^2 = f‖^2 + (2 : ℝ) * t * inner f g + t^2 * g‖^2 :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

√⟪f + t * g, f + t * g_ℝ ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ

√⟪f + t * g, f + t * g_ℝ ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ

√⟪f + t * g, f + t * g_ℝ ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ

f + t * g, f + t * g_ℝ = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ

f + t * g, f + t * g_ℝ = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :

f + t * g, f + t * g_ℝ = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :

f + t * g, f + t * g_ℝ = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:

f + t * g, f + t * g_ℝ = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:

f + t * g, f + t * g_ℝ = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:

a_f_tg set_repr (f + t * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:

a_f_tg set_repr (f + t * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:

a_f_tg set_repr (f + t * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
repr_add: (fun i => a_f i + Set.Nonempty.some i) set_repr (f + t * g)

a_f_tg set_repr (f + t * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:

a_f_tg set_repr (f + t * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
repr_add: (fun i => a_f i + Set.Nonempty.some i) set_repr (f + t * g)
repr_mul: (fun i => t * a_g i) set_repr (t * g)

a_f_tg set_repr (f + t * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:

a_f_tg set_repr (f + t * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
repr_add: (fun i => a_f i + Set.Nonempty.some i) set_repr (f + t * g)
repr_mul: (fun i => t * a_g i) set_repr (t * g)

a_f_tg set_repr (f + t * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
repr_add: (fun i => a_f i + (fun i => t * a_g i) i) set_repr (f + t * g)
repr_mul: (fun i => t * a_g i) set_repr (t * g)

a_f_tg set_repr (f + t * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
repr_add: (fun i => a_f i + (fun i => t * a_g i) i) set_repr (f + t * g)
repr_mul: (fun i => t * a_g i) set_repr (t * g)

a_f_tg set_repr (f + t * g)

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)

f + t * g, f + t * g_ℝ = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)

f + t * g, f + t * g_ℝ = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)

H_inner (f + t * g) (f + t * g) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)

∑' (i : ℕ), v i * a_f_tg i * a_f_tg i = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)

∑' (i : ℕ), v i * a_f_tg i * a_f_tg i = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)

∑' (i : ℕ), v i * a_f_tg i * a_f_tg i = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)

(i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
i:

v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
i:

v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)

(i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)

∑' (i : ℕ), v i * a_f_tg i * a_f_tg i = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
[Meta.Tactic.simp.rewrite] distribute:1000, v i * a_f_tg i * a_f_tg i ==> v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)

∑' (i : ℕ), (v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)

∑' (i : ℕ), (v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)

∑' (i : ℕ), (v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)

∑' (i : ℕ), (v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)

∑' (b : ℕ), (v b * a_f b * a_f b + 2 * t * (v b * a_f b * a_g b)) + ∑' (b : ℕ), t ^ 2 * (v b * Set.Nonempty.some b * Set.Nonempty.some b) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)

∑' (b : ℕ), (v b * a_f b * a_f b + 2 * t * (v b * a_f b * a_g b)) + ∑' (b : ℕ), t ^ 2 * (v b * Set.Nonempty.some b * Set.Nonempty.some b) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)

∑' (b : ℕ), (v b * a_f b * a_f b + 2 * t * (v b * a_f b * a_g b)) + ∑' (b : ℕ), t ^ 2 * (v b * Set.Nonempty.some b * Set.Nonempty.some b) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

H_inner h h = h ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

H_inner h h = h ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

H_inner h h = h ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

h, h_ℝ = h ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

h, h_ℝ = h ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
h: (H v e μ)

h ^ 2 = h ^ 2

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2

∑' (b : ℕ), (v b * a_f b * a_f b + 2 * t * (v b * a_f b * a_g b)) + ∑' (b : ℕ), t ^ 2 * (v b * Set.Nonempty.some b * Set.Nonempty.some b) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2

∑' (b : ℕ), v b * Set.Nonempty.some b * Set.Nonempty.some b + ∑' (b : ℕ), 2 * t * (v b * Set.Nonempty.some b * Set.Nonempty.some b) + ∑' (b : ℕ), t ^ 2 * (v b * Set.Nonempty.some b * Set.Nonempty.some b) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2

∑' (b : ℕ), v b * Set.Nonempty.some b * Set.Nonempty.some b + ∑' (b : ℕ), 2 * t * (v b * Set.Nonempty.some b * Set.Nonempty.some b) + ∑' (b : ℕ), t ^ 2 * (v b * Set.Nonempty.some b * Set.Nonempty.some b) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i

∑' (b : ℕ), v b * Set.Nonempty.some b * Set.Nonempty.some b + ∑' (b : ℕ), 2 * t * (v b * Set.Nonempty.some b * Set.Nonempty.some b) + ∑' (b : ℕ), t ^ 2 * (v b * Set.Nonempty.some b * Set.Nonempty.some b) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i

∑' (b : ℕ), v b * Set.Nonempty.some b * Set.Nonempty.some b + ∑' (b : ℕ), 2 * t * (v b * Set.Nonempty.some b * Set.Nonempty.some b) + ∑' (b : ℕ), t ^ 2 * (v b * Set.Nonempty.some b * Set.Nonempty.some b) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i

∑' (b : ℕ), v b * Set.Nonempty.some b * Set.Nonempty.some b + ∑' (b : ℕ), 2 * t * (v b * Set.Nonempty.some b * Set.Nonempty.some b) + t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i

∑' (b : ℕ), v b * Set.Nonempty.some b * Set.Nonempty.some b + ∑' (b : ℕ), 2 * t * (v b * Set.Nonempty.some b * Set.Nonempty.some b) + ∑' (b : ℕ), t ^ 2 * (v b * Set.Nonempty.some b * Set.Nonempty.some b) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i

f ^ 2 + ∑' (b : ℕ), 2 * t * (v b * Set.Nonempty.some b * Set.Nonempty.some b) + t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i

∑' (b : ℕ), v b * Set.Nonempty.some b * Set.Nonempty.some b + ∑' (b : ℕ), 2 * t * (v b * Set.Nonempty.some b * Set.Nonempty.some b) + ∑' (b : ℕ), t ^ 2 * (v b * Set.Nonempty.some b * Set.Nonempty.some b) = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i

f ^ 2 + ∑' (b : ℕ), 2 * t * (v b * Set.Nonempty.some b * Set.Nonempty.some b) + t ^ 2 * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i

f ^ 2 + ∑' (b : ℕ), 2 * t * (v b * Set.Nonempty.some b * Set.Nonempty.some b) + t ^ 2 * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), v i * a_f i * a_g i

f ^ 2 + ∑' (b : ℕ), 2 * t * (v b * Set.Nonempty.some b * Set.Nonempty.some b) + t ^ 2 * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), v i * a_f i * a_g i

f ^ 2 + ∑' (b : ℕ), 2 * t * (v b * Set.Nonempty.some b * Set.Nonempty.some b) + t ^ 2 * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), v i * a_f i * a_g i

f ^ 2 + 2 * t * ∑' (i : ℕ), v i * a_f i * a_g i + t ^ 2 * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), v i * a_f i * a_g i

f ^ 2 + 2 * t * ∑' (i : ℕ), v i * a_f i * a_g i + t ^ 2 * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), v i * a_f i * a_g i

f ^ 2 + 2 * t * ∑' (i : ℕ), v i * a_f i * a_g i + t ^ 2 * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), v i * a_f i * a_g i

∑' (i : ℕ), v i * a_f i * a_g i = f, g_ℝ

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:

f + t * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), v i * a_f i * a_g i
tsum_to_inner: ∑' (i : ℕ), v i * a_f i * a_g i = f, g_ℝ

f ^ 2 + 2 * t * ∑' (i : ℕ), v i * a_f i * a_g i + t ^ 2 * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
t:
inner_nn: 0 f + t * g, f + t * g_ℝ
a_f:= Set.Nonempty.some :
a_g:= Set.Nonempty.some :
a_f_tg:= fun i => a_f i + t * a_g i:
a_f_tg_repr: a_f_tg set_repr (f + t * g)
distribute: (i : ℕ), v i * a_f_tg i * a_f_tg i = v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i) + t ^ 2 * (v i * a_g i * a_g i)
add_summable: Summable fun i => v i * a_f i * a_f i + 2 * t * (v i * a_f i * a_g i)
tsum_to_norm: (h : (H v e μ)), ∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i = h ^ 2
const_out✝: ∑' (i : ℕ), t ^ 2 * (v i * a_g i * a_g i) = t ^ 2 * ∑' (i : ℕ), v i * a_g i * a_g i
const_out: ∑' (i : ℕ), 2 * t * (v i * a_f i * a_g i) = 2 * t * ∑' (i : ℕ), v i * a_f i * a_g i
tsum_to_inner: ∑' (i : ℕ), v i * a_f i * a_g i = f, g_ℝ

f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2 = f ^ 2 + 2 * t * f, g_ℝ + t ^ 2 * g ^ 2

Goals accomplished! 🐙
lemma H_cauchy_schwarz : inner f g <= f * g :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: ¬‖g 0
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: ¬‖g 0
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

pos
f, g_ℝ f * g

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / (g ^ 2 * g ^ 2) * g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / (g ^ 2 * g ^ 2) * g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / (g ^ 2 * g ^ 2) * g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / g ^ 2 * 1 / g ^ 2 * g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / g ^ 2 * 1 / g ^ 2 * g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / g ^ 2 * 1 / g ^ 2 * g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / g ^ 2 * (1 / g ^ 2 * g ^ 2) = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / g ^ 2 * (1 / g ^ 2 * g ^ 2) = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / g ^ 2 * 1 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / g ^ 2 * 1 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / g ^ 2 * 1 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / g ^ 2 * 1 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + 2 * t₀ * f, g_ℝ + f, g_ℝ ^ 2 / g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + -2 * f, g_ℝ ^ 2 / g ^ 2 + f, g_ℝ ^ 2 / g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀

P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + -2 * f, g_ℝ ^ 2 / g ^ 2 + f, g_ℝ ^ 2 / g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0: P t₀ = f ^ 2 + 2 * t₀ * f, g_ℝ + t₀ ^ 2 * g ^ 2

f ^ 2 + -2 * f, g_ℝ ^ 2 / g ^ 2 + f, g_ℝ ^ 2 / g ^ 2 = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

Goals accomplished! 🐙

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 P t₀
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

pos
f, g_ℝ f * g

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

f, g_ℝ ^ 2 (f * g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

f, g_ℝ ^ 2 (f * g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

f, g_ℝ ^ 2 (f * g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

f, g_ℝ ^ 2 (f * g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

f, g_ℝ ^ 2 (f * g) ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

f, g_ℝ ^ 2 f ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

f, g_ℝ ^ 2 (f * g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

f, g_ℝ ^ 2 f ^ 2 * g ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

f, g_ℝ ^ 2 * (g ^ 2)⁻¹ f ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

f, g_ℝ ^ 2 * (g ^ 2)⁻¹ f ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

f, g_ℝ ^ 2 (f * g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

f, g_ℝ ^ 2 * (g ^ 2)⁻¹ f ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

0 f ^ 2 - f, g_ℝ ^ 2 * (g ^ 2)⁻¹
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2

0 f ^ 2 - f, g_ℝ ^ 2 * (g ^ 2)⁻¹

Goals accomplished! 🐙
--rw [←sq_abs (inner f g)] at sq_ineq
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
sq_ineq: f, g_ℝ ^ 2 (f * g) ^ 2

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
sq_ineq: f, g_ℝ ^ 2 (f * g) ^ 2

pos
f, g_ℝ f * g

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
sq_ineq: f, g_ℝ * f, g_ℝ (f * g) ^ 2

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
sq_ineq: f, g_ℝ * f, g_ℝ (f * g) ^ 2

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
sq_ineq: f, g_ℝ * f, g_ℝ (f * g) ^ 2

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
sq_ineq: f, g_ℝ * f, g_ℝ (f * g) ^ 2

pos
f, g_ℝ f * g

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
sq_ineq: f, g_ℝ * f, g_ℝ f * g * (f * g)

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
sq_ineq: f, g_ℝ * f, g_ℝ f * g * (f * g)

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0
hg_sq: g ^ 2 0
P:= fun t => f + t * g ^ 2:
t₀:= -⟪f, g_ℝ / g ^ 2:
P_nonneg: 0 f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
P_t0_val: P t₀ = f ^ 2 - f, g_ℝ ^ 2 / g ^ 2
sq_ineq: f, g_ℝ * f, g_ℝ f * g * (f * g)
norm_mul_nonneg: 0 f * g

pos
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g 0

pos
f, g_ℝ f * g

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g = 0

neg
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g = 0

neg
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g = 0

neg
f, g_ℝ f * g

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: √⟪g, g_ℝ = 0

neg
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: √⟪g, g_ℝ = 0

neg
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: √⟪g, g_ℝ = 0

neg
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0

neg
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0

neg
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0

neg
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
f, 0_ℝ f * 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
f, 0_ℝ f * 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
f, 0_ℝ f * 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
f, 0_ℝ f * 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
f, 0_ℝ f * 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
f, 0_ℝ f * 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
0 f * 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
0 f * 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
0 f * 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
0 f * 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
0 f * √⟪0, 0_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ f * g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
0 f * √⟪0, 0_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
0 f * 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
hg: g, g_ℝ = 0
g_eq_0: g = 0

neg
0 f * 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f, g_ℝ f * g
[Meta.Tactic.simp.rewrite] Real.sqrt_zero:1000, 0 ==> 0 [Meta.Tactic.simp.rewrite] @mul_zero:1000, f * 0 ==> 0 [Meta.Tactic.simp.rewrite] @le_refl:1000, 0 0 ==> True

Goals accomplished! 🐙
lemma ineq_add_norm : f + g <= f + g :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f + g f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

hb
0 f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
f + g * f + g (f + g) * (f + g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f + g f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

hb
0 f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

hb
0 f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
f + g * f + g (f + g) * (f + g)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f + g f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

h
f + g * f + g (f + g) * (f + g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

h
f + g * f + g (f + g) * (f + g)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

h
f + g ^ 2 (f + g) * (f + g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f + g f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

h
f + g ^ 2 (f + g) * (f + g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

h
f + g ^ 2 (f + g) * (f + g)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f + g f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

h
f + g ^ 2 (f + g) ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

2 * f, g_ℝ 2 * (f * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

2 * f, g_ℝ 2 * (f * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

2 * f, g_ℝ 2 * (f * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
this: f, g_ℝ f * g

2 * f, g_ℝ 2 * (f * g)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

2 * f, g_ℝ 2 * (f * g)

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
cauchy_schwarz: 2 * f, g_ℝ 2 * (f * g)
ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 f ^ 2 + 2 * (f * g) + g ^ 2

f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
cauchy_schwarz: 2 * f, g_ℝ 2 * (f * g)
ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 f ^ 2 + 2 * (f * g) + g ^ 2

f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
cauchy_schwarz: 2 * f, g_ℝ 2 * (f * g)
ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 f ^ 2 + 2 * (f * g) + g ^ 2

f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
cauchy_schwarz: 2 * f, g_ℝ 2 * (f * g)
ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2

f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f + g f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * 1 * f, g_ℝ + 1 ^ 2 * g ^ 2

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f + g f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * 1 * f, g_ℝ + 1 ^ 2 * g ^ 2

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * 1 * f, g_ℝ + 1 ^ 2 * g ^ 2

h
f + g ^ 2 (f + g) ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + 1 ^ 2 * g ^ 2

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + 1 ^ 2 * g ^ 2

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f + g f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + 1 ^ 2 * g ^ 2

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + 1 ^ 2 * g ^ 2

h
f + g ^ 2 (f + g) ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f + g f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f ^ 2 + 2 * f, g_ℝ + g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f + g f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2

h
f + g ^ 2 (f + g) ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2

1 * g = g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2

1 * g = g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2

1 * g = g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2
x: Ω

a.h
(1 * g) x = g x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2

1 * g = g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2
x: Ω

a.h
(1 * g) x = g x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2
x: Ω

a.h
(1 * g) x = g x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2
x: Ω

a.h
1 * g x = g x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2

1 * g = g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2
x: Ω

a.h
1 * g x = g x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2
x: Ω

a.h
1 * g x = g x

Goals accomplished! 🐙

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

f + g f + g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + 1 * g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2
one_mul_g_eq_g: 1 * g = g

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2
one_mul_g_eq_g: 1 * g = g

h
f + g ^ 2 (f + g) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
sq_ineq: f + g ^ 2 (f + g) ^ 2
distrib_norm: f + 1 * g ^ 2 = f ^ 2 + 2 * f, g_ℝ + g ^ 2
one_mul_g_eq_g: 1 * g = g

h
f + g ^ 2 (f + g) ^ 2

Goals accomplished! 🐙
end Inner /- We show properties on the distance induced by the inner product of H. -/ namespace Dist open Inner Ring Group variable {v : eigen} {e : Ω ℝ} : Measure Ω} [IsFiniteMeasure μ] (f g : H v e μ) lemma H_dist_self (a : H v e μ) : dist a a = 0 :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

dist a a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

dist a a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

dist a a = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a - a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

dist a a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a - a = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a - a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a - a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a - a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a - a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a - a = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a + -1 * a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a - a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
(a + -1 * a) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a - a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
(a + -1 * a) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
(a + -1 * a) x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
a x + (-1 * a) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a - a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
a x + (-1 * a) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
a x + (-1 * a) x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
a x + -1 * a x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a - a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
a x + -1 * a x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
a x + -1 * a x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
x: Ω

a.h
a x + -1 * a x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

a - a = 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

dist a a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
a_sub_a_eq_0: a - a = 0

a - a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
a_sub_a_eq_0: a - a = 0

0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
a_sub_a_eq_0: a - a = 0

0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

dist a a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
a_sub_a_eq_0: a - a = 0

0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
a_sub_a_eq_0: a - a = 0

0 = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
a_sub_a_eq_0: a - a = 0

√⟪0, 0_ℝ = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

dist a a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
a_sub_a_eq_0: a - a = 0

√⟪0, 0_ℝ = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
a_sub_a_eq_0: a - a = 0

0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)
a_sub_a_eq_0: a - a = 0

0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a: (H v e μ)

dist a a = 0

Goals accomplished! 🐙
lemma dist_rw (a b : H v e μ) : (dist a b) = Real.sqrt (∑' i, v.1 i * ((set_repr_ne a).some i - (set_repr_ne b).some i)^2) :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

a - b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

a - b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

a - b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

(H_inner (a - b) (a - b)) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

(∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:

(∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:

(∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:

repr set_repr (a - b)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:

repr set_repr (a - b)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:

repr set_repr (a - b)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_minus_b: (fun i => -1 * Set.Nonempty.some i) set_repr (-1 * b)

repr set_repr (a - b)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:

repr set_repr (a - b)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_minus_b: (fun i => -1 * Set.Nonempty.some i) set_repr (-1 * b)
repr_a_sub_b: (fun i => Set.Nonempty.some i + Set.Nonempty.some i) set_repr (a + -1 * b)

repr set_repr (a - b)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:

repr set_repr (a - b)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_minus_b: (fun i => -1 * Set.Nonempty.some i) set_repr (-1 * b)
repr_a_sub_b: (fun i => Set.Nonempty.some i + Set.Nonempty.some i) set_repr (a + -1 * b)

repr set_repr (a - b)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_minus_b: (fun i => -1 * Set.Nonempty.some i) set_repr (-1 * b)
repr_a_sub_b: (fun i => Set.Nonempty.some i + (fun i => -1 * Set.Nonempty.some i) i) set_repr (a + -1 * b)

repr set_repr (a - b)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_minus_b: (fun i => -1 * Set.Nonempty.some i) set_repr (-1 * b)
repr_a_sub_b: (fun i => Set.Nonempty.some i + (fun i => -1 * Set.Nonempty.some i) i) set_repr (a + -1 * b)

repr set_repr (a - b)

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)

(∑' (i : ℕ), v i * Set.Nonempty.some i * Set.Nonempty.some i) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)

(∑' (i : ℕ), v i * repr i * repr i) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)

(∑' (i : ℕ), v i * repr i * repr i) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :

(∑' (i : ℕ), v i * repr i * repr i) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :

(∑' (i : ℕ), v i * repr i * repr i) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :

(∑' (i : ℕ), v i * repr i * repr i) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
[Meta.Tactic.simp.rewrite] show i, v.1 i * repr i * repr i = v.1 i * (repr i) ^ 2 by intro i; ring:1000, v i * repr i * repr i ==> v i * repr i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :

(∑' (i : ℕ), v i * repr i * repr i) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
[Meta.Tactic.simp.rewrite] show i, v.1 i * repr i * repr i = v.1 i * (repr i) ^ 2 by intro i; ring:1000, v i * repr i * repr i ==> v i * repr i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :
i:

v i * repr i * repr i = v i * repr i ^ 2
[Meta.Tactic.simp.rewrite] show i, v.1 i * repr i * repr i = v.1 i * (repr i) ^ 2 by intro i; ring:1000, v i * repr i * repr i ==> v i * repr i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :
i:

v i * repr i * repr i = v i * repr i ^ 2
[Meta.Tactic.simp.rewrite] show i, v.1 i * repr i * repr i = v.1 i * (repr i) ^ 2 by intro i; ring:1000, v i * repr i * repr i ==> v i * repr i ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :

(i : ℕ), v i * repr i * repr i = v i * repr i ^ 2
[Meta.Tactic.simp.rewrite] show i, v.1 i * repr i * repr i = v.1 i * (repr i) ^ 2 by intro i; ring:1000, v i * repr i * repr i ==> v i * repr i ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :

(∑' (i : ℕ), v i * repr i ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :

(∑' (i : ℕ), v i * repr i ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
[Meta.Tactic.simp.rewrite] show i, v.1 i * (repr i) ^ 2 = v.1 i * (a_r i - b_r i) ^ 2 by intro i; ring:1000, v i * repr i ^ 2 ==> v i * (a_r i - b_r i) ^ 2 [Meta.Tactic.simp.rewrite] @eq_self:1000, (∑' (i : ℕ), v i * (a_r i - b_r i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) ==> True
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :

(∑' (i : ℕ), v i * repr i ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
[Meta.Tactic.simp.rewrite] show i, v.1 i * (repr i) ^ 2 = v.1 i * (a_r i - b_r i) ^ 2 by intro i; ring:1000, v i * repr i ^ 2 ==> v i * (a_r i - b_r i) ^ 2 [Meta.Tactic.simp.rewrite] @eq_self:1000, (∑' (i : ℕ), v i * (a_r i - b_r i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) ==> True
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :
i:

v i * repr i ^ 2 = v i * (a_r i - b_r i) ^ 2
[Meta.Tactic.simp.rewrite] show i, v.1 i * (repr i) ^ 2 = v.1 i * (a_r i - b_r i) ^ 2 by intro i; ring:1000, v i * repr i ^ 2 ==> v i * (a_r i - b_r i) ^ 2 [Meta.Tactic.simp.rewrite] @eq_self:1000, (∑' (i : ℕ), v i * (a_r i - b_r i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) ==> True
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :
i:

v i * repr i ^ 2 = v i * (a_r i - b_r i) ^ 2
[Meta.Tactic.simp.rewrite] show i, v.1 i * (repr i) ^ 2 = v.1 i * (a_r i - b_r i) ^ 2 by intro i; ring:1000, v i * repr i ^ 2 ==> v i * (a_r i - b_r i) ^ 2 [Meta.Tactic.simp.rewrite] @eq_self:1000, (∑' (i : ℕ), v i * (a_r i - b_r i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) ==> True
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
repr:= fun i => Set.Nonempty.some i + -1 * Set.Nonempty.some i:
repr_a_minus_b: repr set_repr (a - b)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :

(i : ℕ), v i * repr i ^ 2 = v i * (a_r i - b_r i) ^ 2
[Meta.Tactic.simp.rewrite] show i, v.1 i * (repr i) ^ 2 = v.1 i * (a_r i - b_r i) ^ 2 by intro i; ring:1000, v i * repr i ^ 2 ==> v i * (a_r i - b_r i) ^ 2 [Meta.Tactic.simp.rewrite] @eq_self:1000, (∑' (i : ℕ), v i * (a_r i - b_r i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) ==> True

Goals accomplished! 🐙

Goals accomplished! 🐙
lemma H_dist_comm (a b : H v e μ) : dist a b = dist b a :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = dist b a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = dist b a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

(∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) = dist b a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = dist b a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

(∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

(∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = dist b a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
a_r:= Set.Nonempty.some :

(∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = dist b a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :

(∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = dist b a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :

(∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
[Meta.Tactic.simp.rewrite] show i, v.1 i * (a_r i - b_r i) ^ 2 = v.1 i * (b_r i - a_r i) ^ 2 by intro i; ring:1000, v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2 ==> v i * (b_r i - a_r i) ^ 2 [Meta.Tactic.simp.rewrite] @eq_self:1000, (∑' (i : ℕ), v i * (b_r i - a_r i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) ==> True
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :

(∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2)
[Meta.Tactic.simp.rewrite] show i, v.1 i * (a_r i - b_r i) ^ 2 = v.1 i * (b_r i - a_r i) ^ 2 by intro i; ring:1000, v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2 ==> v i * (b_r i - a_r i) ^ 2 [Meta.Tactic.simp.rewrite] @eq_self:1000, (∑' (i : ℕ), v i * (b_r i - a_r i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) ==> True
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :
i:

v i * (a_r i - b_r i) ^ 2 = v i * (b_r i - a_r i) ^ 2
[Meta.Tactic.simp.rewrite] show i, v.1 i * (a_r i - b_r i) ^ 2 = v.1 i * (b_r i - a_r i) ^ 2 by intro i; ring:1000, v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2 ==> v i * (b_r i - a_r i) ^ 2 [Meta.Tactic.simp.rewrite] @eq_self:1000, (∑' (i : ℕ), v i * (b_r i - a_r i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) ==> True
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :
i:

v i * (a_r i - b_r i) ^ 2 = v i * (b_r i - a_r i) ^ 2
[Meta.Tactic.simp.rewrite] show i, v.1 i * (a_r i - b_r i) ^ 2 = v.1 i * (b_r i - a_r i) ^ 2 by intro i; ring:1000, v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2 ==> v i * (b_r i - a_r i) ^ 2 [Meta.Tactic.simp.rewrite] @eq_self:1000, (∑' (i : ℕ), v i * (b_r i - a_r i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) ==> True
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
a_r:= Set.Nonempty.some :
b_r:= Set.Nonempty.some :

(i : ℕ), v i * (a_r i - b_r i) ^ 2 = v i * (b_r i - a_r i) ^ 2
[Meta.Tactic.simp.rewrite] show i, v.1 i * (a_r i - b_r i) ^ 2 = v.1 i * (b_r i - a_r i) ^ 2 by intro i; ring:1000, v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2 ==> v i * (b_r i - a_r i) ^ 2 [Meta.Tactic.simp.rewrite] @eq_self:1000, (∑' (i : ℕ), v i * (b_r i - a_r i) ^ 2) = (∑' (i : ℕ), v i * (Set.Nonempty.some i - Set.Nonempty.some i) ^ 2) ==> True

Goals accomplished! 🐙

Goals accomplished! 🐙
lemma H_dist_nonneg (a b : H v e μ) : 0 <= dist a b :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

0 dist a b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

0 dist a b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

0 dist a b

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

0 a - b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

0 dist a b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

0 a - b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

0 a - b

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

0 (H_inner (a - b) (a - b))
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

0 dist a b

Goals accomplished! 🐙
lemma H_eq_of_dist_eq_zero {a b : H v e μ} : dist a b = 0 a = b :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: dist a b = 0

a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: dist a b = 0

a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: dist a b = 0

a = b

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b = 0

a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b = 0

a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b = 0

a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b = 0

a = b

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: √⟪a - b, a - b_ℝ = 0

a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: √⟪a - b, a - b_ℝ = 0

a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: √⟪a - b, a - b_ℝ = 0

a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0

a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0

a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0

a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0

a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: Ω

a.h
a x = b x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: Ω

a.h
a x = b x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: Ω

a x - b x = 0 a x = b x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: Ω
h: a x - b x = 0

a x = b x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: Ω
h: a x - b x = 0

a x = b x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: Ω

a x - b x = 0 a x = b x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x

a.h
a x - b x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x

a.h
a x - b x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x

a.h
a x - b x = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x

a.h
a x + -1 * b x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x

a.h
a x + -1 * b x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a - b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x

a.h
a x + -1 * b x = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x

a.h
a x + -1 * b x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x

a.h
a x + -1 * b x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x
a_minus_b_eq_zero_val: (a + -1 * b) = 0

a.h
a x + -1 * b x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x
a_minus_b_eq_zero_val: (a + -1 * b) = 0

a.h
a x + -1 * b x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x
a_minus_b_eq_zero_val: (a + -1 * b) = 0

a.h
a x + -1 * b x = 0

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x
a_minus_b_eq_zero_val: (fun x => a x + -1 * b x) = 0

a.h
a x + -1 * b x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x
a_minus_b_eq_zero_val: (fun x => a x + -1 * b x) = 0

a.h
a x + -1 * b x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x
a_minus_b_eq_zero_val: (fun x => a x + -1 * b x) = 0

a.h
a x + -1 * b x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x
a_minus_b_eq_zero_val: (fun x => a x + -1 * b x) = 0

a.h
0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)
zero_dist: a - b, a - b_ℝ = 0
a_minus_b_eq_zero: a + -1 * b = 0
x: Ω
dif_imp_eq: a x - b x = 0 a x = b x
a_minus_b_eq_zero_val: (fun x => a x + -1 * b x) = 0

a.h
0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b: (H v e μ)

dist a b = 0 a = b

Goals accomplished! 🐙
lemma H_dist_triangle (a b c : H v e μ) : dist a c dist a b + dist b c :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

dist a c dist a b + dist b c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

dist a c dist a b + dist b c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

dist a c dist a b + dist b c

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c dist a b + dist b c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

dist a c dist a b + dist b c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c dist a b + dist b c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c dist a b + dist b c

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c a - b + dist b c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

dist a c dist a b + dist b c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c a - b + dist b c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c a - b + dist b c

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c a - b + b - c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

dist a c dist a b + dist b c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c a - b + b - c

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c = a - b + (b - c)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c = a - b + (b - c)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c = a - b + (b - c)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = (a - b + (b - c)) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c = a - b + (b - c)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = (a - b + (b - c)) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = (a - b + (b - c)) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = (a - b) x + (b - c) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c = a - b + (b - c)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = (a - b) x + (b - c) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = (a - b) x + (b - c) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = a x + -1 * b x + (b - c) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c = a - b + (b - c)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = a x + -1 * b x + (b - c) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = a x + -1 * b x + (b - c) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = a x + -1 * b x + (b x + -1 * c x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c = a - b + (b - c)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = a x + -1 * b x + (b x + -1 * c x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = a x + -1 * b x + (b x + -1 * c x)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = a x + -1 * c x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

a - c = a - b + (b - c)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = a x + -1 * c x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
x: Ω

a.h
(a - c) x = a x + -1 * c x

Goals accomplished! 🐙

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

dist a c dist a b + dist b c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
split_a_sub_c: a - c = a - b + (b - c)

a - c a - b + b - c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
split_a_sub_c: a - c = a - b + (b - c)

a - b + (b - c) a - b + b - c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)
split_a_sub_c: a - c = a - b + (b - c)

a - b + (b - c) a - b + b - c
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, a, b, c: (H v e μ)

dist a c dist a b + dist b c

Goals accomplished! 🐙
end Dist /- - We instanciate the `NormedAddCommGroup` and `InnerProductSpace ℝ` typeclasses for H. -/ variable {v : eigen} {e : Ω ℝ} : Measure Ω} [IsFiniteMeasure μ] lemma coe_mul_nat_fun (f : H v e μ) (n : ℕ) : (λ (n : ℕ) (f : H v e μ) (n : ℝ) * f) n f = (n : ℝ) * f :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:

(fun n f => n * f) n f = n * f

Goals accomplished! 🐙
lemma coe_mul_int_fun (f : H v e μ) (z : ℤ) : (λ (z : ℤ) (f : H v e μ) (z : ℝ) * f) z f = (z : ℝ) * f :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
z:

(fun z f => z * f) z f = z * f

Goals accomplished! 🐙
lemma cast_nat_succ (n : ℕ) : (n + 1) = (n : ℝ) + 1 :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:

(n + 1) = n + 1
[Meta.Tactic.simp.rewrite] @Nat.cast_add:1000, (n + 1) ==> n + 1 [Meta.Tactic.simp.rewrite] @Nat.cast_one:1000, 1 ==> 1 [Meta.Tactic.simp.rewrite] @eq_self:1000, n + 1 = n + 1 ==> True

Goals accomplished! 🐙
lemma mul_succ_eq (f : H v e μ) (n : ℕ) : ((n : ℝ) + 1) * f = ((n : ℝ) * f + f) :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:

(n + 1) * f = n * f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:
x: Ω

a.h
((n + 1) * f) x = (n * f + f) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:

(n + 1) * f = n * f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:
x: Ω
n':= n:

a.h
((n + 1) * f) x = (n * f + f) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:

(n + 1) * f = n * f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:
x: Ω
n':= n:

a.h
((n + 1) * f) x = (n * f + f) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:
x: Ω
n':= n:

a.h
((n + 1) * f) x = (n * f + f) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:
x: Ω
n':= n:

a.h
((n + 1) * f) x = (n' * f) x + f x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:

(n + 1) * f = n * f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:
x: Ω
n':= n:

a.h
((n + 1) * f) x = (n' * f) x + f x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:
x: Ω
n':= n:

a.h
((n + 1) * f) x = (n' * f) x + f x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:
x: Ω
n':= n:

a.h
((n + 1) * f) x = n' * f x + f x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:

(n + 1) * f = n * f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:
x: Ω
n':= n:

a.h
((n + 1) * f) x = n' * f x + f x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:
x: Ω
n':= n:

a.h
((n + 1) * f) x = n' * f x + f x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:
x: Ω
n':= n:

a.h
(n' + 1) * f x = n' * f x + f x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
n:

(n + 1) * f = n * f + f

Goals accomplished! 🐙
noncomputable instance : NormedAddCommGroup (H v e μ) where dist := λ f g dist f g edist := λ f g ENNReal.ofReal (dist f g) norm := λ f norm f add := λ f g f + g add_assoc := Group.H_add_assoc zero_add := Group.H_zero_add add_zero := Group.H_add_zero nsmul := λ n f (n : ℝ) * f neg := λ f -f zsmul := λ z f (z : ℝ) * f add_left_neg := Group.H_add_left_neg add_comm := Group.H_add_comm dist_self := Dist.H_dist_self dist_comm := Dist.H_dist_comm dist_triangle := Dist.H_dist_triangle edist_dist := λ f g rfl eq_of_dist_eq_zero := Dist.H_eq_of_dist_eq_zero dist_eq := λ x y

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
x, y: (H v e μ)

dist x y = x - y

Goals accomplished! 🐙
nsmul_zero :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), (fun n f => n * f) 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), (fun n f => n * f) 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), (fun n f => n * f) 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

(fun n f => n * f) 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), (fun n f => n * f) 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

(fun n f => n * f) 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

0 * f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

0 * f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), (fun n f => n * f) 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

0 * f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

0 * f = 0
[Meta.Tactic.simp.rewrite] CharP.cast_eq_zero:1000, 0 ==> 0 [Meta.Tactic.simp.rewrite] @eq_self:1000, 0 = 0 ==> True

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

0 * f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), (fun n f => n * f) 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
(0 * f) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), (fun n f => n * f) 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
(0 * f) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
(0 * f) x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
0 * f x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), (fun n f => n * f) 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
0 * f x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
0 * f x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
0 * f x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), (fun n f => n * f) 0 x = 0

Goals accomplished! 🐙

Goals accomplished! 🐙
nsmul_succ :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (x : (H v e μ)), (fun n f => n * f) (n + 1) x = (fun n f => n * f) n x + x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (x : (H v e μ)), (fun n f => n * f) (n + 1) x = (fun n f => n * f) n x + x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (x : (H v e μ)), (fun n f => n * f) (n + 1) x = (fun n f => n * f) n x + x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun n f => n * f) (n + 1) f = (fun n f => n * f) n f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (x : (H v e μ)), (fun n f => n * f) (n + 1) x = (fun n f => n * f) n x + x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun n f => n * f) (n + 1) f = (fun n f => n * f) n f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = (fun n f => n * f) n f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun n f => n * f) (n + 1) f = (fun n f => n * f) n f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = (fun n f => n * f) n f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun n f => n * f) (n + 1) f = (fun n f => n * f) n f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = n * f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = n * f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (x : (H v e μ)), (fun n f => n * f) (n + 1) x = (fun n f => n * f) n x + x

Goals accomplished! 🐙

Goals accomplished! 🐙
zsmul_zero' :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : (H v e μ)), (fun z f => z * f) 0 a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : (H v e μ)), (fun z f => z * f) 0 a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : (H v e μ)), (fun z f => z * f) 0 a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

(fun z f => z * f) 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : (H v e μ)), (fun z f => z * f) 0 a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

(fun z f => z * f) 0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

0 * f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

0 * f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : (H v e μ)), (fun z f => z * f) 0 a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

0 * f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

0 * f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

0 * f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : (H v e μ)), (fun z f => z * f) 0 a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
(0 * f) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : (H v e μ)), (fun z f => z * f) 0 a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
(0 * f) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
(0 * f) x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
0 * f x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : (H v e μ)), (fun z f => z * f) 0 a = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
0 * f x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
0 * f x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
0 * f x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : (H v e μ)), (fun z f => z * f) 0 a = 0

Goals accomplished! 🐙

Goals accomplished! 🐙
zsmul_succ' :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => z * f) (Int.ofNat n) a + a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => z * f) (Int.ofNat n) a + a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => z * f) (Int.ofNat n) a + a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (Int.ofNat (Nat.succ n)) f = (fun z f => z * f) (Int.ofNat n) f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => z * f) (Int.ofNat n) a + a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (Int.ofNat (Nat.succ n)) f = (fun z f => z * f) (Int.ofNat n) f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (Int.ofNat (Nat.succ n)) f = (fun z f => z * f) (Int.ofNat n) f + f

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (Int.ofNat (n + 1)) f = (fun z f => z * f) (Int.ofNat n) f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => z * f) (Int.ofNat n) a + a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (Int.ofNat (n + 1)) f = (fun z f => z * f) (Int.ofNat n) f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(Int.ofNat (n + 1)) * f = (fun z f => z * f) (Int.ofNat n) f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(Int.ofNat (n + 1)) * f = (fun z f => z * f) (Int.ofNat n) f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => z * f) (Int.ofNat n) a + a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(Int.ofNat (n + 1)) * f = (fun z f => z * f) (Int.ofNat n) f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(Int.ofNat (n + 1)) * f = (fun z f => z * f) (Int.ofNat n) f + f

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = (fun z f => z * f) (Int.ofNat n) f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => z * f) (Int.ofNat n) a + a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = (fun z f => z * f) (Int.ofNat n) f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = (fun z f => z * f) (Int.ofNat n) f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = (fun z f => z * f) (Int.ofNat n) f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => z * f) (Int.ofNat n) a + a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = (fun z f => z * f) (Int.ofNat n) f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = (Int.ofNat n) * f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = (Int.ofNat n) * f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => z * f) (Int.ofNat n) a + a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = (Int.ofNat n) * f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = (Int.ofNat n) * f + f

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(n + 1) * f = n * f + f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.ofNat (Nat.succ n)) a = (fun z f => z * f) (Int.ofNat n) a + a

Goals accomplished! 🐙

Goals accomplished! 🐙
zsmul_neg' :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (Int.negSucc n) f = -(fun z f => z * f) ((Nat.succ n)) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (Int.negSucc n) f = -(fun z f => z * f) ((Nat.succ n)) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (Int.negSucc n) f = -(fun z f => z * f) ((Nat.succ n)) f

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (-(n + 1)) f = -(fun z f => z * f) ((Nat.succ n)) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (-(n + 1)) f = -(fun z f => z * f) ((Nat.succ n)) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (-(n + 1)) f = -(fun z f => z * f) ((Nat.succ n)) f

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (-(n + 1)) f = -(fun z f => z * f) ((n + 1)) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(fun z f => z * f) (-(n + 1)) f = -(fun z f => z * f) ((n + 1)) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -(fun z f => z * f) ((n + 1)) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -(fun z f => z * f) ((n + 1)) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -(fun z f => z * f) ((n + 1)) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -(fun z f => z * f) ((n + 1)) f

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -1 * (fun z f => z * f) ((n + 1)) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -1 * (fun z f => z * f) ((n + 1)) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -1 * (fun z f => z * f) ((n + 1)) f

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -1 * (fun z f => z * f) (n + 1) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -1 * (fun z f => z * f) (n + 1) f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -1 * ((n + 1) * f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -1 * ((n + 1) * f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -1 * ((n + 1) * f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -1 * ((n + 1) * f)
[Meta.Tactic.simp.rewrite] @Int.cast_add:1000, (n + 1) ==> ↑↑n + 1 [Meta.Tactic.simp.rewrite] @Int.cast_natCast:10000, ↑↑n ==> n [Meta.Tactic.simp.rewrite] @Int.cast_one:1000, 1 ==> 1 [Meta.Tactic.simp.rewrite] @eq_self:1000, n + 1 = n + 1 ==> True

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -1 * ((n + 1) * f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -1 * ((n + 1) * f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

(-(n + 1)) * f = -1 * ((n + 1) * f)
[Meta.Tactic.simp.rewrite] @neg_add_rev:1000, -(n + 1) ==> -1 + -↑n [Meta.Tactic.simp.rewrite] @Int.cast_add:1000, (-1 + -↑n) ==> (-1) + (-↑n) [Meta.Tactic.simp.rewrite] @Int.cast_neg:1000, (-1) ==> -↑1 [Meta.Tactic.simp.rewrite] @Int.cast_one:1000, 1 ==> 1 [Meta.Tactic.simp.rewrite] @Int.cast_neg:1000, (-↑n) ==> -↑↑n [Meta.Tactic.simp.rewrite] @Int.cast_natCast:10000, ↑↑n ==> n [Meta.Tactic.simp.rewrite] @neg_add_rev:1000, -(n + 1) ==> -1 + -↑n [Meta.Tactic.simp.rewrite] @eq_self:1000, -1 + -↑n = -1 + -↑n ==> True

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)

-(n + 1) * f = -1 * ((n + 1) * f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)
x: Ω

a.h
(-(n + 1) * f) x = (-1 * ((n + 1) * f)) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)
x: Ω

a.h
(-(n + 1) * f) x = (-1 * ((n + 1) * f)) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)
x: Ω

a.h
(-(n + 1) * f) x = (-1 * ((n + 1) * f)) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)
x: Ω

a.h
-(n + 1) * f x = (-1 * ((n + 1) * f)) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)
x: Ω

a.h
-(n + 1) * f x = (-1 * ((n + 1) * f)) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)
x: Ω

a.h
-(n + 1) * f x = (-1 * ((n + 1) * f)) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)
x: Ω

a.h
-(n + 1) * f x = -1 * ((n + 1) * f) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)
x: Ω

a.h
-(n + 1) * f x = -1 * ((n + 1) * f) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)
x: Ω

a.h
-(n + 1) * f x = -1 * ((n + 1) * f) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
n:
f: (H v e μ)
x: Ω

a.h
-(n + 1) * f x = -1 * ((n + 1) * f x)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(n : ℕ) (a : (H v e μ)), (fun z f => z * f) (Int.negSucc n) a = -(fun z f => z * f) ((Nat.succ n)) a

Goals accomplished! 🐙

Goals accomplished! 🐙
open Inner noncomputable instance : InnerProductSpace (H v e μ) where smul := λ a f a * f one_smul :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(b : (H v e μ)), 1 b = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(b : (H v e μ)), 1 b = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(b : (H v e μ)), 1 b = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
b: (H v e μ)

1 b = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(b : (H v e μ)), 1 b = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
b: (H v e μ)
x: Ω

a.h
(1 b) x = b x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(b : (H v e μ)), 1 b = b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
b: (H v e μ)
x: Ω

a.h
(1 b) x = b x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
b: (H v e μ)
x: Ω

a.h
(1 b) x = b x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
b: (H v e μ)
x: Ω

a.h
1 * b x = b x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(b : (H v e μ)), 1 b = b

Goals accomplished! 🐙

Goals accomplished! 🐙
mul_smul :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : ℝ) (b : (H v e μ)), (x * y) b = x y b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : ℝ) (b : (H v e μ)), (x * y) b = x y b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : ℝ) (b : (H v e μ)), (x * y) b = x y b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
x, y:
b: (H v e μ)

(x * y) b = x y b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : ℝ) (b : (H v e μ)), (x * y) b = x y b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e✝: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
x, y:
b: (H v e μ)
e: Ω

a.h
((x * y) b) e = (x y b) e
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : ℝ) (b : (H v e μ)), (x * y) b = x y b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e✝: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
x, y:
b: (H v e μ)
e: Ω

a.h
((x * y) b) e = (x y b) e
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e✝: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
x, y:
b: (H v e μ)
e: Ω

a.h
((x * y) b) e = (x y b) e

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e✝: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
x, y:
b: (H v e μ)
e: Ω

a.h
x * y * b e = (x y b) e
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : ℝ) (b : (H v e μ)), (x * y) b = x y b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e✝: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
x, y:
b: (H v e μ)
e: Ω

a.h
x * y * b e = (x y b) e
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e✝: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
x, y:
b: (H v e μ)
e: Ω

a.h
x * y * b e = (x y b) e

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e✝: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
x, y:
b: (H v e μ)
e: Ω

a.h
x * y * b e = x * y b e
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : ℝ) (b : (H v e μ)), (x * y) b = x y b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e✝: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
x, y:
b: (H v e μ)
e: Ω

a.h
x * y * b e = x * y b e
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e✝: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
x, y:
b: (H v e μ)
e: Ω

a.h
x * y * b e = x * y b e

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e✝: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
x, y:
b: (H v e μ)
e: Ω

a.h
x * y * b e = x * (y * b e)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : ℝ) (b : (H v e μ)), (x * y) b = x y b

Goals accomplished! 🐙

Goals accomplished! 🐙
smul_zero :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ), a 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ), a 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ), a 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:

a 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ), a 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
x: Ω

a.h
(a 0) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ), a 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
x: Ω

a.h
(a 0) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
x: Ω

a.h
(a 0) x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
x: Ω

a.h
a * 0 = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ), a 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
x: Ω

a.h
a * 0 = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
x: Ω

a.h
a * 0 = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
x: Ω

a.h
a * 0 = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ), a 0 = 0

Goals accomplished! 🐙

Goals accomplished! 🐙
smul_add :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (x y : (H v e μ)), a (x + y) = a x + a y
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (x y : (H v e μ)), a (x + y) = a x + a y
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (x y : (H v e μ)), a (x + y) = a x + a y
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
f, g: (H v e μ)

a (f + g) = a f + a g
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (x y : (H v e μ)), a (x + y) = a x + a y
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
f, g: (H v e μ)
x: Ω

a.h
(a (f + g)) x = (a f + a g) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (x y : (H v e μ)), a (x + y) = a x + a y
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
f, g: (H v e μ)
x: Ω

a.h
(a (f + g)) x = (a f + a g) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
f, g: (H v e μ)
x: Ω

a.h
(a (f + g)) x = (a f + a g) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
f, g: (H v e μ)
x: Ω

a.h
a * (f + g) x = (a f + a g) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (x y : (H v e μ)), a (x + y) = a x + a y
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
f, g: (H v e μ)
x: Ω

a.h
a * (f + g) x = (a f + a g) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
f, g: (H v e μ)
x: Ω

a.h
a * (f + g) x = (a f + a g) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
f, g: (H v e μ)
x: Ω

a.h
a * (f x + g x) = (a f + a g) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (x y : (H v e μ)), a (x + y) = a x + a y
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
f, g: (H v e μ)
x: Ω

a.h
a * (f x + g x) = (a f + a g) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
f, g: (H v e μ)
x: Ω

a.h
a * (f x + g x) = (a f + a g) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
a:
f, g: (H v e μ)
x: Ω

a.h
a * (f x + g x) = a * f x + a * g x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (x y : (H v e μ)), a (x + y) = a x + a y

Goals accomplished! 🐙

Goals accomplished! 🐙
add_smul :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(r s : ℝ) (x : (H v e μ)), (r + s) x = r x + s x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(r s : ℝ) (x : (H v e μ)), (r + s) x = r x + s x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(r s : ℝ) (x : (H v e μ)), (r + s) x = r x + s x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r, s:
f: (H v e μ)

(r + s) f = r f + s f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(r s : ℝ) (x : (H v e μ)), (r + s) x = r x + s x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r, s:
f: (H v e μ)
x: Ω

a.h
((r + s) f) x = (r f + s f) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(r s : ℝ) (x : (H v e μ)), (r + s) x = r x + s x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r, s:
f: (H v e μ)
x: Ω

a.h
((r + s) f) x = (r f + s f) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r, s:
f: (H v e μ)
x: Ω

a.h
((r + s) f) x = (r f + s f) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r, s:
f: (H v e μ)
x: Ω

a.h
(r + s) * f x = (r f + s f) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(r s : ℝ) (x : (H v e μ)), (r + s) x = r x + s x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r, s:
f: (H v e μ)
x: Ω

a.h
(r + s) * f x = (r f + s f) x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r, s:
f: (H v e μ)
x: Ω

a.h
(r + s) * f x = (r f + s f) x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r, s:
f: (H v e μ)
x: Ω

a.h
(r + s) * f x = r * f x + s * f x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(r s : ℝ) (x : (H v e μ)), (r + s) x = r x + s x

Goals accomplished! 🐙

Goals accomplished! 🐙
zero_smul :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

0 f = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
(0 f) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
(0 f) x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
(0 f) x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
0 * f x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), 0 x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
0 * f x = 0 x
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
0 * f x = 0 x

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)
x: Ω

a.h
0 * f x = 0
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), 0 x = 0

Goals accomplished! 🐙

Goals accomplished! 🐙
norm_smul_le :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

r f r * f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r f = r * f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r f = r * f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r f = r * f

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f = r * f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f = r * f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f = r * f

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f = |r| * f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f = |r| * f
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f * r * f = |r| * f * (|r| * f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f * r * f = |r| * f * (|r| * f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f * r * f = |r| * f * (|r| * f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f * r * f = |r| * f * (|r| * f)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f ^ 2 = |r| * f * (|r| * f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f ^ 2 = |r| * f * (|r| * f)
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f ^ 2 = |r| * f * (|r| * f)

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f ^ 2 = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f ^ 2 = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f, r * f_ℝ = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f ^ 2 = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f, r * f_ℝ = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f, r * f_ℝ = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f, r * f_ℝ = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * r * f, f_ℝ = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f, r * f_ℝ = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * (r * f, f_ℝ) = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * f, r * f_ℝ = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * (r * f ^ 2) = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * (r * f ^ 2) = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * (r * f ^ 2) = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * (r * f ^ 2) = (|r| * f) ^ 2

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r ^ 2 * f ^ 2 = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
r * (r * f ^ 2) = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
|r| ^ 2 * f ^ 2 = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
r:
f: (H v e μ)

a
|r| ^ 2 * f ^ 2 = (|r| * f) ^ 2
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(a : ℝ) (b : (H v e μ)), a b a * b

Goals accomplished! 🐙

Goals accomplished! 🐙
inner := H_inner norm_sq_eq_inner :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), x ^ 2 = RCLike.re x, x_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), x ^ 2 = RCLike.re x, x_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), x ^ 2 = RCLike.re x, x_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

f ^ 2 = RCLike.re f, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), x ^ 2 = RCLike.re x, x_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

f ^ 2 = RCLike.re f, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

f, f_ℝ = RCLike.re f, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f: (H v e μ)

f, f_ℝ = RCLike.re f, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x : (H v e μ)), x ^ 2 = RCLike.re x, x_ℝ

Goals accomplished! 🐙

Goals accomplished! 🐙
conj_symm :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : (H v e μ)), (starRingEnd ℝ) y, x_ℝ = x, y_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : (H v e μ)), (starRingEnd ℝ) y, x_ℝ = x, y_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : (H v e μ)), (starRingEnd ℝ) y, x_ℝ = x, y_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(starRingEnd ℝ) g, f_ℝ = f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : (H v e μ)), (starRingEnd ℝ) y, x_ℝ = x, y_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(starRingEnd ℝ) g, f_ℝ = f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(starRingEnd ℝ) g, f_ℝ = g, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)

(starRingEnd ℝ) g, f_ℝ = g, f_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : (H v e μ)), (starRingEnd ℝ) y, x_ℝ = x, y_ℝ

Goals accomplished! 🐙

Goals accomplished! 🐙
add_left :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y z : (H v e μ)), x + y, z_ℝ = x, z_ℝ + y, z_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y z : (H v e μ)), x + y, z_ℝ = x, z_ℝ + y, z_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y z : (H v e μ)), x + y, z_ℝ = x, z_ℝ + y, z_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g, h: (H v e μ)

f + g, h_ℝ = f, h_ℝ + g, h_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y z : (H v e μ)), x + y, z_ℝ = x, z_ℝ + y, z_ℝ

Goals accomplished! 🐙

Goals accomplished! 🐙
smul_left :=

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : (H v e μ)) (r : ℝ), r x, y_ℝ = (starRingEnd ℝ) r * x, y_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : (H v e μ)) (r : ℝ), r x, y_ℝ = (starRingEnd ℝ) r * x, y_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : (H v e μ)) (r : ℝ), r x, y_ℝ = (starRingEnd ℝ) r * x, y_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
r:

r f, g_ℝ = (starRingEnd ℝ) r * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : (H v e μ)) (r : ℝ), r x, y_ℝ = (starRingEnd ℝ) r * x, y_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
r:

r f, g_ℝ = (starRingEnd ℝ) r * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
r:

r f, g_ℝ = (starRingEnd ℝ) r * f, g_ℝ

Goals accomplished! 🐙
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ
f, g: (H v e μ)
r:

r * f, g_ℝ = (starRingEnd ℝ) r * f, g_ℝ
d:
Ω: Set (Vector d)
inst✝¹: MeasureSpace Ω
v: eigen
e: Ω
μ: Measure Ω
inst✝: IsFiniteMeasure μ

(x y : (H v e μ)) (r : ℝ), r x, y_ℝ = (starRingEnd ℝ) r * x, y_ℝ

Goals accomplished! 🐙

Goals accomplished! 🐙